UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


GIFT  OF 

CARNEGIE   INSTITUTION 
OF  WASHINGTON 


EXPERIMENTS  WITH  THE 
DISPLACEMENT  INTERFEROMETER 


By  CARL  BARUS 

Hazard  Professor  of  Physics  and  Dean  of  the  Graduate  Department 
in  Brown  University 


WASHINGTON,  D.  C. 

Published  by  the  Carnegie  Institution  of  Washington 


43 


EXPERIMENTS  WITH  THE 
DISPLACEMENT  INTERFEROMETER 


BY  CARL  BARUS 

Hazard  Professor  of  Physics  and  Dean  of  the  Graduate  Department 
in  Brown  University 


WASHINGTON,  D.  C. 

Published  by  the  Carnegie  Institution  of  Washington 
1915 


CARNEGIE  INSTITUTION  OF  WASHINGTON 
PUBLICATION  No.  229 


PREFACE. 

The  present  volume  contains  applications  of  the  displacement  interfer- 
ometer to  subjects  which  suggested  themselves  from  time  to  time.  Unfor- 
tunately it  was  not  possible,  in  the  laboratory  of  Brown  University,  which  is 
situated  on  a  hill  in  the  middle  of  a  large  city,  to  carry  out  any  experiment 
to  its  final  degree  of  rigor.  Quiet  surroundings,  a  location  free  from  tremor, 
and  irregular  temperature  variations  would  have  been  necessary.  But  the 
development  of  methods  of  the  kind  in  question  was  nevertheless  quite  feasi- 
ble ;  and  without  attempting  to  push  them  to  a  limit,  the  range  of  application 
could  be  fully  investigated. 

Among  the  subjects  selected  for  treatment  was  the  horizontal  pendulum. 
In  the  first  part  of  Chapter  I  certain  available  forms  of  the  pendulum,  with 
and  without  a  float,  are  considered  and  tested  as  to  their  discrepancies, 
through  long  lapses  of  time,  by  a  reflection  method.  Among  the  interesting 
results  obtained  is  the  suggestion  of  an  apparatus  capable  of  measuring 
changes  of  elongation  to  the  amount  of  even  less  than  4Xio~10  of  the  total 
length  per  vanishing  interference  ring. 

In  the  second  part  of  the  chapter  the  interferometer  itself  is  used,  a  service- 
able method  of  application  worked  out,  and  the  range  of  application  studied 
through  many  months.  With  a  relatively  very  wide  scope  (several  seconds 
of  arc)  there  should  be  no  difficulty,  under  proper  surroundings,  of  measuring 
changes  of  inclination  as  small  as  3  X  io~4  seconds  of  arc  per  interference  ring, 
and  it  is  probable  that  one  could  reach  smaller  angles  by  modifying  parts  of 
the  pendulum. 

In  Chapter  II  an  attempt  is  made  to  use  this  interferential  horizontal  pen- 
dulum for  the  measurement  of  the  gravitational  attraction  of  two  parallel 
disks.  What  was  obtained,  however,  was  a  definite  repulsion  of  the  disks, 
decreasing  with  their  distance  apart  and  appreciable  even  within  1.5  mm.  of 
this  distance.  As  the  method  of  measurement  contemplates  the  viscosity  of 
the  film  of  air  between  the  disks,  and  as  the  effect  of  any  natural  charge  or 
potential  would  be  insignificant  in  comparison  with  the  forces  observed,  it  is 
probable  that  the  repulsion  in  question  is  attributable  to  the  molecular 
atmospheres  by  which  the  disks  are  surrounded  in  air,  supposing  that  such 
atmospheres  of  gas  increase  in  density  as  the  surface  of  the  disk  is  approached. 

Chapter  III  is  introduced  as  a  severe  test  on  the  interference  equation 
employed  for  the  case  of  path  differences  resulting  when  glass  columns  as 
much  as  10  inches  long  are  inserted  in  one  of  the  component  beams  of  the 
displacement  interferometer.  It  appears  that  the  constants  of  any  dispersion 
formula  may  be  obtained  directly  from  these  observations.  The  equations 
for  the  relations  of  displacement  and  wave-length  increments  show,  however, 
that  the  anticipation  of  great  precision  in  the  determination  of  refraction, 

iii 

209210 


iv  PREFACE. 

by  lengthening  the  column  of  glass,  is  not  fulfilled.    The  ellipses  become  pro- 
portionately more  sluggish  in  their  motion  as  the  path  difference  is  increased. 

In  Chapter  IV  a  number  of  incidental  experiments,  on  allied  subjects,  have 
been  grouped  together.  In  the  first  paper  the  possible  bearing  of  certain  disk 
colors  of  circular  gratings  on  the  somewhat  similar  phenomenon  in  coronas 
is  discussed.  The  second  paper  deals  with  the  performance  of  the  easily 
available  film  grating  to  replace  the  ruled-glass  grating,  for  purposes  of  dis- 
placement interferometry,  from  a  practical  standpoint.  With  the  same  end 
in  view  the  third  paper  considers  the  use  of  the  Nernst  filament  as  an  available 
illuminator,  in  the  absence  of  the  arc  lamp  or  sunlight.  In  conclusion,  an 
interesting  case  of  regular  reflection  and  refraction  of  scattered  light,  bearing 
on  the  X-ray  phenomena  recently  discovered  by  Professor  Bragg,  is  treated 
in  the  fourth  paper. 

In  Chapter  V,  finally,  following  the  suggestive  experiments  made  in  an 
earlier  report,  the  displacement  interferometer  is  directly  applied  to  the 
quadrant  electrometer.  In  the  several  hundreds  of  adjustments  made  no 
serious  difficulty  was  encountered  in  the  optical  parts  of  the  experiments, 
and  that  was  the  question  chiefly  at  issue.  The  sensitiveness  obtained  in 
this  way  should  have  been  of  the  order  of  a  millionth  of  a  volt  per  vanishing 
interference  ring;  but  owing  to  the  uninterrupted  commotions  surrounding 
the  laboratory  already  referred  to,  possibly  also  to  difficulties  residing  in  the 
electrometer,  this  limit  could  not  be  reached.  The  experiments,  therefore, 
largely  explore  the  available  scope  of  the  method. 

My  thanks  are  due  to  Mrs.  D.  T.  Knight  and  to  Miss  R.  R.  Snow  for 
efficient  assistance  in  connection  with  the  preparation  of  the  papers  for  the 
press. 

CARL  BARUS. 
BROWN  UNIVERSITY,  September  25, 1915. 


CONTENTS. 


CHAPTER  I. — Elliptic  Intcrferometry  Applied  to  the  Horizontal  Pendulum. 
PART  I.— Direct  Differential  Reflection. 

PAOB 

1.  Introduction.     Method.    Fig.  i I 

2.  Equations.    Figs.  2,3,4 2 

3.  Observations.    Fig.  5 6 

4.  New  apparatus,  without  float.    Figs.  6,7 7 

5.  Observations.    Fig.  8 9 

6.  New  apparatus,  with  float.    Horizontal  pivots II 

7.  Observations.    Figs.  9,  100,  lob 12 

8.  Second  apparatus,  with  float.    Jeweled  bearings.    Fig.  na,  nb,  lie 14 

9.  Observations.    Fig.  12 15 

10.  Observations,  continued.    Fig.  13 17 

11.  Effect  of  temperature  on  the  float,  etc 18 

12.  Further  observations.    Fig.  14 20 

13.  Effect  of  temperature  on  the  scaffolding.    Fig.  15 21 

14.  Inferences.    Fig.  16 24 

15.  The  precision  measurement  of  elongations.    Fig.  17 25 

16.  Improved  pendulum 28 

17.  Observations  with  the  new  pendulum.    Fig.  18 28 

PART  II. — An  Application  of  the  Displacement  Interferometer  to  the  Horizontal 
Pendulum. 

18.  Introductory 30 

19.  Apparatus.    Figs.  19,  20,  21 31 

20.  Equations 34 

21 .  Observations  with  a  grating  rotating  on  a  fixed  vertical  axis 38 

22.  Observations  with  the  interferometer.     Fig.  22 39 

23.  Further  observations.    Film  grating.    Oil  damper.    Figs.  23A,  236,  24,  25,  26 41 

24.  Inferences 42 

25.  Improved  aluminum  pendulum.    Observations.    Fig.  27 45 

CHAPTER  II. — The  Repulsion  of  Two  Metallic  Disks,  Nearly  in  Contact. 

26.  Apparatus.    Figs.  28,  29 49 

27.  Equations 50 

28.  Equations  for  the  vertical  pendulum.    Fig.  30 51 

29.  Observations  with  small  plates.    Figs.  3iA,  318,  3iC,  32A,  328 52 

30.  Observations.    Plates  of  larger  area.    Tablet.    Figs.  33A,  338,  33C,  330 53 

31.  The  same,  continued.    Metallic  contact.    Table  2.    Figs.  34A,  348 55 

32.  Retardation  due  to  viscosity  of  air.    Table  3.    Fig.  35 57 

33.  Observations,  continued.    Presence  and  absence  of  electrical  contact.     Table  4. 

Fig.  36 60 

34.  Observations,  continued.     Change  of  distance  apart.    Table  5.    Figs.  37,  38,  39A, 

39B 61 

35.  Observations.    Long  periods  and  inversion.    Table  6.    Fig.  40 65 

36.  Plates  electrically  charged.    Tables  7,  8.    Figs.  4iA,  418 67 

37.  Conclusion 70 

CHAPTER  III. —  The  Refraction  of  Long  Glass  Columns  Measured  by  Displacement 
Interferometry. 

38.  Introductory 72 

39.  Glass  columns.    Fig.  42 72 

40.  Equations 74 

41 .  Equations.    Sensitiveness  in  terms  of  displacement 75 

42.  Equations.    Sensitiveness  in  terms  of  order 76 

43.  Observations.    Green  glass  column.    Table  9 77 

44.  Observations.    Blue  glass  column.    Table  10 78 

45.  Observations.    Shorter  column.    Table  1 1 79 

46.  Summary.    Fig.  43 80 

v 


CONTENTS. 

CHAPTER  IV. — Miscellaneous  Papers. 
PART  I. — Experiments  Bearing  on  the  Properties  of  Coronas. 


PAQ! 


47.  Introductory.    Figs.  44,  450,  456,  45c 81 

48.  Experiments  with  a  grating.    Table  12.    Figs.  46,  47 82 

49.  Experiments  with  small  coronas 85 

50.  Experiments    with  large  coronas,  annular  source.    Coronas  by  reflection.    Figs. 

48,  49 86 

51.  Coronas  from  a  point  source.     Figs.  50,  51 88 

PART  II. — Displacement  Interferometry  with  Film  Grating. 

52.  Introductory.    Figs.  52,  53 92 

53.  Films  between  glass  plates 92 

54.  Continued.    The  groups  i  +4,  I  +5.     Figs.  54,  55,  56,  57,  58,  59 93 

55.  Continued.    The  groups  2+4,  2+5 94 

56.  Continued.    The  groups  3+4,  3+5 95 

57.  Centers  of  ellipses 95 

58.  Film  or  ruling  on  one  side  of  the  glass  plate.    Ruled  grating 96 

59.  Continued.    Film  grating  not  cemented  to  glass.    Fig.  60 97 

60.  Single  plate,  film  grating 97 

PART  III. — Elliptic  Interferometry  with  a  Nernst  Filament. 

61.  Introduction 98 

62.  The  Nernst  burner.    Fig.  61 99 

63.  Remarks 99 

PART  IV. — Scattering  in  the  Case  of  Regular  Reflection  from  a  Transparent  Grating, 
an  Analogy  to  the  Reflection  of  X-rays  from  Crystals. 

64.  The  phenomenon.    Fig.  62 100 

65.  Explanation 101 

66.  A  further  analogy  to  the  reflection  of  X-rays 102 

CHAPTER  V. — Displacement  Interferometry  Applied  to  the  Quadrant  Electrometer. 

67.  Apparatus.    Fig.  63 103 

68.  Observations 104 

69.  Observations,  continued.    Fig.  64 105 

70.  Observations,  continued.    Figs.  6sA,  658 105 

71.  Observations,  continued 106 

72.  Further  observations.    Table  13.    Fig.  66 107 

73.  Observations,  continued.    Tables  14,  15,  16,  17,  18 109 

74.  Summary 112 


CHAPTER  I. 


ELLIPTIC  INTERFEROMETRY  APPLIED  TO  THE  HORIZONTAL  PENDULUM. 

PART  I —DIRECT  DIFFERENTIAL  REFLECTION. 

1.  Introduction.  Method.— Before  using  interferences,  it  seemed  inter- 
esting to  apply  the  interferometer  adjustment  to  the  case  of  simple  reflection, 
the  mutual  displacement  of  the  two  direct  images  from  the  front  and  rear 
face  of  the  mirror  on  the  pendulum  being  used  for  the  measurement  of  the 
angle  of  deviation  of  the  pendulum.  In  such  reflection  from  a  glass  plate 
there  is  necessarily  considerable  loss  of  light;  but  at  radii  of  20  and  30  meters, 
when  the  source  of  light  is  a  slit  closely  in  front  of  a  Nernst  filament,  this 
difficulty  is  not  prohibitive. 
It  is  necessary,  however,  that 
the  lens  of  the  collimator,  as 
well  as  the  plates  of  glass  and 
mirror  used,  be  of  high  opti- 
cal quality;  otherwise  it  is 
impossible  to  obtain  sharp 
condensation  at  a  very  dis- 
tant focus.  One  may  also 
concentrate  the  slit  images  to 
a  point  by  a  cylindrical  lens, 
placed  with  its  linear  element 
at  right  angles  to  the  direc- 
tion of  the  slit. 

The  interesting  feature  of 
the  method  is  that  it  is  inde- 
pendent of  any  zero-point,  as 
the  distance  apart  of  the  two  images  on  the  far  screen  at  once  measures  the 
inclination  of  the  pendulum  axis,  the  normal  position  being  that  in  which 
the  two  images  coincide.  If  the  two  opaque  mirrors  are  rigidly  fixed,  the 
direct  or  incident  beam  of  light  from  the  source  and  the  subsidiary  reflecting 
mirrors  may  shift  without  modifying  the  datum  for  the  inclination.  Further- 
more, the  sensitiveness  is  twice  that  of  the  case  of  single  reflection,  other  things 
being  equal.  The  method  is  thus  particularly  adapted  for  the  measurement 
of  the  inclination  of  the  plumb-bob  relatively  to  the  earth. 

The  annexed  diagram,  fig.  i,  will  make  the  method  clear.  Here  5  is  the 
fine  slit  in  front  of  the  Nernst  filament/,  and  /  the  condensing  doublet  (about 
60  cm.  focal  distance;  for  rough  work  an  ordinary  spectacle-glass  answers 
very  well)  of  the  collimator.  It  is  necessary  that  this  lens  be  wide,  weak, 
and  good  in  order  that  a  sharp  focus  and  as  little  loss  of  light  as  practicable 
be  obtained  at  the  distant  focus. 

1 


FIG.  i. 


2        EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

The  pencil  from  the  collimator  strikes  the  plate  of  glass  G  at  the  end  of 
the  horizontal  pendulum,  the  greater  part  of  this,  d',  being  transmitted  to 
the  opaque  mirror  M ,  the  remainder,  d,  reflected  from  the  opaque  mirror  N. 
It  is  advantageous  to  have  M  and  N  equidistant  from  G,  as  nearly  as  prac- 
ticable (20  or  30  cm.),  and  far  from  the  lamp/,  to  avoid  the  menace  of  tem- 
perature discrepancies.  If  the  mounting  is  of  gas-pipe,  water  circulation 
might  be  used,  but  this  is  not  necessary. 

In  the  diagram  the  pencils  d  and  d'  are  normally  reflected  at  M  and  N. 
On  returning  d  is  transmitted  and  d'  reflected,  so  that  the  beams  reunite  and 
proceed  together  to  the  far  focus  F,  20  or  30  meters  distant,  where  they  are 
caught  on  a  paper  or  ground-glass  screen,  or  directly  observed  with  the  lens. 
It  is  particularly  necessary  that  the  movable  reflector  at  G  be  an  excellent 
optical  plate,  i  or  more  inches  square.  When  the  plate  at  G  (which  is  at 
the  extremity  of  the  horizontal  pendulum)  rotates  over  a  small  angle  $,  the 
reflected  rays  d"  and  d'"  now  diverge  in  opposed  directions  from  the  center 
C  on  the  face  of  the  opaque  mirror  N,  and  pass  to  the  distant  foci  F"  and  F'", 
where  they  are  now  at  a  distance  x  apart.  If  the  rotation  of  the  horizontal 
pendulum  were  —  9,  the  positions  of  the  beams  would  be  exchanged  (see  F' 
and  Fiv).  In  other  words,  if  the  pendulum  vibrates,  the  two  foci  F"  and  F'" 
move  in  opposed  directions,  passing  through  each  other,  when  the  normal 
position  is  instantaneously  assumed,  irrespective  of  the  amplitude  of  vibra- 
tion. It  may  be  noted  that  a  similar  adjustment  may  often  with  advantage 
be  attached  to  any  ordinary  pendulum. 

The  mounting  of  the  plate  G  and  the  mirrors  M,  N,  etc.,  is  identical  with 
that  of  the  interferometer  described  in  the  next  section  (the  plate  G  is  there 
replaced  by  a  transparent  grating)  and  need  not  be  treated  here.  Necessarily 
the  collimator  and  the  mirrors  M  and  N  are  attached  to  the  same  pier  which 
carries  the  horizontal  pendulum,  to  the  end  of  which  the  mirror  G  is  attached; 
but  the  horizontal  pendulum  and  its  case  must  otherwise  be  quite  independent 
of  the  goniometric  apparatus. 

2.  Equations. — It  will  be  convenient  to  suppose  the  foci  F,  F",  F'"  to  lie 
in  the  plane  of  the  mirror  M,  and  the  two  mirrors  M  and  N  to  be  equidistant, 
so  that  d=d',  taken  from  the  plate  G  as  an  ideal  plane.  Let  a  be  the  angle 
between  the  normal  to  N  and  the  direction  of  the  incident  pencil;  i.e.,  let  a 
be  the  angle  between  the  mirrors,  made  as  small  and  as  convenient  as  prac- 
ticable. Then  if  the  mirror  at  G  rotates  over  any  angle  0  =  b/2,  the  distance 
apart  x  of  the  foci  F"  and  F'"  will  be 

(i)  x=d     s{n2b     (       '        -f  -  M 

cos  (a— b)  \cosa-f6      cos6/ 

This  equation  may  be  transformed  to 

/2)  .j^__  Lf        cos2  6          ,i        cos  b 


cos  2a+cos  26       2   cos  (a— 6) 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


Since  the  angles  a  and  b  are  invariably  small,  the  cosines  may  be  expanded, 
so  that 


(3) 

is  nearly  true,  or 

(4) 


=2d  tan  b 


Since  b  =  2d  is  exceedingly  small  (but  a  few  seconds), 


(5) 


Finally,  if  a  is  also  sufficiently  small,  which  will  usually  be  the  case,  and  D 
is  2d  or  zd' ,  so  that  the  distances  to  the  far  screen,  F"  and  F"'t  are  measured 
from  the  mirror  N, 
(6)  x=>4D6 

One  may  note  in  passing  that  the  distance  over  which  the  N  ray  travels  from 
coincidence  is 


cos 


(i 
cos  a 


whereas  the  Af  ray  travels  over 

xt=d'  (tan  (a+6)  -tan  a) 

where  *=#i+#2.  Hence,  for  small  angles,  the  N  ray  travels  over  3  times 
the  distance  db  of  the  M  ray,  the  total  being  ^db.  Thus  the  angle  of  devia- 
tion 6  is  measured  by  x,  apart  from  any  other  consideration,  except  that  the 
distance  D  is  very  large  and  therefore  invariable  and  the  sensitiveness  is 
twice  as  large  as  in  the  case  of  single  reflection. 

To  test  this  result  in  its  practical  aspects,  a  millimeter  micrometer  was 
installed  at  the  end  of  the  pendulum,  at  a  distance  of  51.5  cm.  from  the  axis. 
The  two  images  traveled  in  opposite  directions,  in  steps,  from  end  to  end  of  a 
30  cm.  scale,  while  the  micrometer  was  moved  forward  i  mm.,  successively, 
eight  times,  as  follows: 


Micrometer. 

Mean. 

* 

Micrometer. 

Mean. 

X 

o.o  cm. 
.1 

,2 

•3 
•4 

1  6.  i  cm. 
15-7 
15-4 

I5'o 

14.8 

32.3  cm. 
24.7 
17-3 
9-9 
+2.5 

0.5  cm. 

•7 
.8 

14.5  cm. 
14.2 
13-9 
13-7 

-  4-9  cm. 
-12.3 
-19.7 
-27.0 

With  the  exception  of  the  first  and  last  readings  the  steps  of  x  are  7.4  cm. 
and  equidistant.    Hence 

»-•  1/51. 5 -7.4/4^ 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


whence  D  is  equal  to  914  cm.,  which  agreed  with  the  direct  measurement. 
The  center  of  the  images  ("mean")  holds  pretty  well  to  the  scale,  shifting 
but  from  13.7  to  16.1,  while  the  distance  D  is  smaller  and  the  total  angle 
(0.8/51.5  radian,  about  i  degree)  larger  than  would  usually  be  employed. 

In  the  experiments  made  below,  the  distance  D  was  frequently  above  2,000 
cm.  Since  i  second  of  arc  is  about  5X10"*  radians,  the  deflection  x  corre- 
sponding to  6=  i  sec.  would  therefore  be 

*  =  4X20ooXsXio-fl=4Xio-lcm. 

or  nearly  half  a  millimeter.  A  sharp  focus  F",  F"e  is  thus  nevertheless  needed 
if  single  seconds  of  6  are  to  be  read  off  visually.  I  frequently  made  use 
of  what  seemed  to  be  the  internal  diffraction  patterns 
of  the  slit,  fine  bright  lines  in  each  being  used  for 
measurement. 

The  angle  6,  denoting  the  deviation  of  the  pendulum, 
is  invariably  very  large  as  compared  with  the  angle  a, 
the  corresponding  change  of  inclination  of  the  pier  to 
the  plumb-line.  In  fact,  fig.  2 ,  cdg  denotes  the  horizontal 
pendulum,  with  the  grating  at  g,  pivots  at  c  and  d,  the 
center  of  gravity  at  G,  at  a  distance  h  from  the  axis  cd. 
The  latter  prolonged  intersects  the  plumb-line  through 
G  at  e,  all  in  the  plane  of  the  diagram.  The  angle 
between  the  axis  de  and  the  vertical  df  is  <p  in  the  same 
plane.  When  the  axis,  owing  to  the  tilt  of  the  pier, 
takes  a  new  position  de',  the  arc  ee'  is  nearly 

FIG.  2. 


When  <p  is  very  small,  as  is  necessarily  the  case,  h  =  hf  and  H=H'  very 
nearly,  so  that 

hd=Ha  and  h—H<p    whence  a  =  <f>6 

Thus  if  de  is  a  rigid  stick  pivoted  at  d  and  fe  a  flexible  inextensible  line,  the 
motion  is  such  as  if  the  whole  mass  of  the  pendulum  were  concentrated  at  e, 
the  diagram  being  the  plane  of  the  couple  Mgh=MgH<p. 

As  in  a  =  <p6  all  angles  are  given  in  radians,  if  the  angle  <p  is  of  the  order  of 
i°,  the  ratio  a/6  is  but  0.0175.  I  need  merely  instance,  therefore,  if  v?=o.oi, 
since  X  =  4Da/<p  and  D  =  2,000  cm.  (as  above,  conditions  all  of  which  are 
easily  realized),  that  per  second  of  arc  of  a,  #  =  4  cm. 

The  form  which  the  scaffolding  eventually  took  is  shown  in  the  diagram, 
fig.  3,  in  elevation.  All  rods  were  of  K-inch  iron  pipe;  so  that,  if  desirable, 
a  current  of  water  could  have  been  passed  through  the  essential  braces.  FF' 
is  a  long  rectangle  of  gas-pipe,  240  cm.  from  end  to  end  and  10  cm.  high.  Its 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.        5 

direct  attachment  to  the  pier  PP  is  at  b  and  /.  The  ends  of  FF'  are  braced 
by  the  rods  FD  and  F'D  in  a  vertical  plane,  and  by  rods  at  B  (horizontal) 
and  C  (oblique).  The  lower  abutment  of  C  is  about  a  meter  down  toward 
the  rear,  so  that  D,  F',  C,  B  is  a  large  tetrahedron.  This  arrangement  is  at 
the  same  time  adequately  simple  and  firm,  but  it  is  not  of  course  proof  against 
tremors.  In  fact,  not  a  method  was  found  by  which  these  could  be  excluded 
entirely.  They  exist  in  the  pier.  The  optical  parts  are  now  attached  to  either 
of  the  horizontal  rods  FF'  by  strong  clamps  of  the  usual  type  (reentrant 
wedges).  From  the  lamp  at  A,  which  may  be  either  an  arc  or  a  Nernst  fila- 


71 


FIGS.  3  and  4. 

ment,  the  light  passes  successively  through  the  micrometer  slit  S,  the  colli- 
mating  lens  L,  to  the  vertical  plate  of  glass  H  on  the  horizontal  pendulum. 
Thence  it  is  reflected  to  the  opaque  mirror  N  about  20  cm.  behind  the  diagram, 
and  transmitted  to  the  opposite  mirror  M.  From  both  mirrors  it  is  returned 
to  the  plate  of  glass  at  H,  after  which  the  nearly  coincident  reflected  and  trans- 
mitted beams  pass  to  the  left  of  the  diagram  to  the  far  distant  screen  (2,000 
cm.)  on  which  they  are  caught  and  their  distance  apart,  x,  measured. 

The  method  best  suited  for  visual  observation,  which  alone  is  here  at- 
tempted, consisted  in  adjusting  a  clear  glass  millimeter  scale  (fig.  4)  ss,  about 
1 5  cm.  long,  seen  distinctly  through  the  lens  /,  and  noting  the  position  of  the 
arriving  beams  of  light,  m  and  n,  practically  in  focus  on  55.  A  dark  box  open 
at  both  ends  B  surrounds  the  beams  and  I  is  moved  on  a  slide. 

The  opaque  mirrors  (here  plane)  are  necessarily  adjustable  around  hori- 
zontal and  vertical  axes  and  the  micrometer  slit  must  be  very  fine.  At  a  is 
a  fine  adjustment  (horizontal  and  vertical  axes)  for  the  mirror  M,  though  it 
need  only  be  used  in  the  interferometry  below.  The  gas-pipes,  when  partly 
screwed,  partly  clamped,  together,  make  a  very  serviceable  framework  for 
experimental  purposes. 


6        EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

For  convenience  in  observation,  it  is  necessary  that  the  horizontal  pendu- 
lum be  damped.  A  water  damper,  as  well  as  an  oil  damper,  was  used  without 
apparent  disadvantage,  care  being  taken  to  keep  the  long  copper  trough  in 
which  the  vane  dips  full  of  water.  Soldered  parts  must  be  covered  with  a 
varnish,  for  instance  of  wax  and  resin,  as  there  will  otherwise  be  slow  galvanic 
corrosion  and  a  precipitate. 

3.  Observations.  —  The  observations  were  carried  out  for  experimental  pur- 
poses only,  because  an  adequate  lens  and  plate-glass  mirrors  were  not  at  hand, 
and  because  the  hill  on  which  the  laboratory  is  built  is  in  a  continual  state 
of  tremor,  due  to  the  heavy  car  freightage,  both  on  the  surface  of  the  hill 
and  through  it.  The  pier,  moreover,  was  not  protected  and  insulated  to  an 
extent  needed  in  refined  seismological  work.  Hence  the  chief  purpose  of 
these  experiments  is  to  indicate  the  deviations  to  be  expected  prior  to  the 
interferometer  work  of  the  next  section.  In  the  preliminary  experiments, 
the  pendulum  was  used  without  a  damper  and  changes  of  inclination  of  the 
horizontal  pendulum  of  a  =  0.4  second  per  day,  or  even  i  second  in  several 
successive  days,  were  not  infrequent.  On  other  days  the  pendulum  was  rela- 
tively fixed.  The  succession  of  points  was  quite  regular,  and  maxima  and 
minima  frequent. 

Observations  of  a  more  definite  character  were  taken  between  September 
26  and  December  3,  1913.  They  were  computed  throughout  and  charted. 
They,  however,  have  little  more  than  the  local  interest  specified,  and  I  will 
therefore  merely  give  an  example  of  part  of  the  results  in  the  graph,  fig.  5. 
By  equation  (6)  above,  where  x  is  the  distance  apart  of  the  two  images  of 
the  slit  and  D  the  intervening  space  between  the  plane  of  these  images  and 
the  fixed  mirror  N  nearest  the  lamp, 


d  being  the  angular  deflection  of  the  pendulum.    Since  D  =  2,000  cm., 
6=  125X10-**  radians. 

Furthermore,  a  =  <pd,  where  <p  is  the  inclination  of  the  line  drawn  through  the 
points  of  the  pivots  to  the  vertical,  and  a  the  change  of  inclination  of  the 
pier  to  the  vertical  corresponding  to  9.  Hence, 


The  value  of  <p  found  below  in  the  work  with  the  interferometer  is  <p=  0.0108 
radian,  so  that  a  is  a  little  larger  than  i  per  cent  of  d.    Thus 

«  =  i  •  3  5  X  io-*x  radians  =  -  28*  second,  nearly. 

In  fig.  5,  the  values  of  a  in  seconds  are  given  from  November  4  to  November 
26.  The  apparatus  was  interfered  with  from  time  to  time,  as  shown  at  a, 
and  new  modifications  were  introduced.  The  water  damper  was  used  through- 
out, so  that  the  pendulum  did  not  vibrate. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.        7 

The  curve  as  a  whole  represents  the  contortions  of  the  pier,  probably  while 
it  was  being  gradually  dried  out  from  its  moist  condition  of  the  summer  by 
the  steam  heat  radiating  from  the  steam-pipes  in  the  room.  This  was  partic- 
ularly apparent  toward  the  end  of  the  curve,  at  a  time  when  the  room  for 
incidental  reasons  happened  to  be  excessively  hot.  But  a  continual  increase 
of  a  is  apparent,  showing  that  the  structure  as  a  whole  was  gradually  tipping 
in  one  direction.  Blasting  operations  were  in  progress  in  a  tunnel  underneath 
the  hill,  but  it  is  not  probable,  judging  from  later  results,  that  these  affect 


24       Z6 


the  readings  of  the  horizontal  pendulum  in  the  lapse  of  time.  It  is  impossible 
to  come  to  definite  conclusions  at  present,  but  it  is  not  out  of  the  question 
that  actual  seasonal  changes  in  the  hill  itself  have  also  been  recorded.  Thus 
the  occurrence  of  a  gale  always  produces  a  marked  temporary  effect.  I  have 
not,  however,  thought  it  worth  while  to  compare  the  graph  of  fig.  5  with  other 
graphs  (temperature,  etc.)  which  were  simultaneously  taken,  as  such  work, 
to  be  trustworthy,  must  be  done  in  this  laboratory  in  the  summer  months, 
and  it  will  then  be  advantageous  to  use  the  interferometer,  as  shown  below. 

4.  New  apparatus,  without  float. — To  mount  the  symmetrical  form  of 
pendulum  described  below,  an  iron  scaffolding  was  installed  (in  the  absence 
of  a  suitable  pier),  erected  on  the  cement  foundation  layer  of  the  physical 
laboratory.  The  truss  sustaining  the  optical  parts  is  shown  in  perspective 
in  fig.  6  and  the  horizontal  pendulum  independent  of  this,  i.e.,  free  from  it, 
in  fig.  7.  In  practice  the  apparatus,  fig.  6,  surrounded  fig.  7.  The  feet  BA, 
BC,  B'A',  B'C  of  the  framework  are  bolted  to  the  firm  layer  of  cement  at 
AC  and  A'C'  and  carry  the  horizontal  rods  GH,  which  with  DI  make  a  paral- 
lelogram. FGE  is  the  rod  for  mounting  the  optical  parts,  secured  by  the 
braces  DC  and  DEcaFD  in  a  vertical  plane.  Lateral  braces  EH,  El,  FH, 


8        EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

FI,  A'J,  C'K  provide  for  firmness  in  the  horizontal  direction.  The  feet 
ABC  and  A'B'C  were  additionally  cross-braced  (not  shown).  This  truss  is 
not  ideal;  but  as  it  had  to  pass  through  a  square  hole  in  the  floor  of  the  room, 


& 


FIG.  6. 

50  cm.  above  the  cement  floor,  conditions  had  to  be  compromised.  It  seemed, 
however,  to  meet  the  initial  requirements  of  the  experiment  adequately.  All 
rods  are  of  ><-inch  gas-pipe,  usually  screwed  fast  at  one  end  and  clamped  at 


FIG.  7. 

the  other.    The  position  of  the  horizontal  pendulum  on  a  separate  mounting 

is  sketched  in  at  kPh'P',  the  glass  plate  being  at  h  and  h',  the  pivots  at  p  and  pf. 

The  two  adjustable  opaque  mirrors  are  shown  at  M  and  N,  the  former  being 

about  20  cm.  to  the  rear,  and  held  by  clamps.    5  is  the  slit  and  L  the  lens  of 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.        9 

the  collimator,  the  lamp  being  at  A,  on  a  separate  stand.  In  fig.  7,  the  tall 
standard  A  ABB  of  i-inch  brass  pipe,  well  braced  (not  shown),  supports  both 
the  case  abcf  of  tin  plate  and  the  pivot  supports  de  of  the  horizontal  pendulum 
HPH'P'.  The  rear  and  sides  of  the  case  are  rigidly  fixed,  but  the  front  may 
be  removed  as  a  whole.  Similarly,  the  square  boxes  abcf,  of  which  a  and  c 
are  provided  with  glass  plates,  slide  out  horizontally  or  vertically,  both  in 
front  and  in  the  rear.  The  pendulum  is  thus  easily  accessible  for  adjustment. 
The  pivots  may  be  revolved  around  a  horizontal  axis,  or  moved  fore  and  aft, 
right  and  left,  or  up  and  down.  The  fore-and-aft  movement  is  provided 
with  a  screw  adjustment,  like  the  right-and-left  movement.  The  method 
of  attachment  is  much  the  same  as  that  to  be  described  below. 

Pendulum  and  case  are  quite  independent  of  the  truss  (fig.  6) ,  a  very  essen- 
tial condition,  as  the  truss  must  often  be  touched  for  optical  adjustment.  In 
a  later  adjustment  the  case  also  was  mounted  in  complete  independence  of 
the  pendulum. 

The  present  symmetrical  horizontal  pendulum  was  made  of  X~mcn  alumi- 
num tubing,  the  vertical  brace  PP  of  ^-inch  aluminum  tubing.  The  junc- 
tions are  brass  tubing  and  the  cups  and  slots  for  pivots  either  jeweled  or  of 
glass-hard  steel.  There  is  a  slot  for  the  d  pivot  and  a  conical  hollow  for  the 
e  pivot.  Inasmuch  as  the  horizontal  pendulum  is  invariably  under  tremor, 
with  the  consequent  absence  of  static  friction,  the  pivot  e  in  the  first  experi- 
ment was  adjusted  vertically,  though  the  means  were  at  hand  for  adjusting 
it  in  any  inclined  direction,  as  will  presently  be  shown.  The  horizontal  brace 
gh  is  of  tense  brass  wire,  forming  a  rhombus  when  seen  from  above,  the  object 
being  to  enhance  the  lateral  rigidity  of  the  pendulum.  Finally,  the  plane 
parallel  glass  plates,  H,  H',  lie  to  the  front  of  the  pendulum,  so  that  a  beam 
of  light  may  pass  across,  parallel  to  its  plane,  as  called  for  in  some  of  the 
interferometer  measurements.  They  are  below  the  line  of  horizontal  symmetry 
and  each  is  adjustable  around  a  horizontal  and  a  vertical  axis. 

The  case  at  /  was  specially  adapted  for  the  installation  of  a  float  K,  which 
will  be  described  presently  and  was  not  used  in  the  first  experiments.  The 
vat  C  of  the  float  must  be  supported  on  an  entirely  separate  standard  (not 
shown). 

5.  Observations. — The  total  weight  of  the  apparatus,  including  the  mirrors, 
was  M—  740  grams,  with  the  center  of  gravity  at  h=  13.9  cm.  from  the  axis; 
the  effective  length  of  the  arm  was  R  =  59.5  cm.  and  the  period  T=2g  seconds. 

The  moment  of  inertia  for  an  axis  through  the  center  of  gravity  was  found 
to  be  1.51  X  io6.  Since  the  mass  was  740  grams,  this  is  equivalent  to  a  radius 
of  gyration  4*0  =  45.2  cm.  Hence,  since  the  distance  of  the  center  of  gravity 
from  the  pivotal  axis  is  ^  =  13.9  cm.,  the  radius  of  gyration  for  the  same  axis 
will  be  *  =  47-3  cm.  From  this  and  the  above  period, 

47T2*2 

(f>=*      =  7 -7 1  Xio-3  radians, 


10     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

or  about  0.44  degree.  Hence,  the  change  of  inclination  of  the  pendulum  is 
nearly  a  =  0.00770.  The  force  at  a  distance  #  =  59.5  cm.  from  the  pivotal 
axis  (place  of  the  plate  of  glass  or  grating)  is 


supposing  the  interferometer  to  be  used  and  that  AJV  is  the  displacement 
of  the  micrometer.  Thus  FR  here  and  in  the  preceding  case,  when  referred 
to  the  same  <p  and  AN,  shows 

F/<p&N=*  1080  and  4000,  respectively, 

so  that  the  present  horizontal  pendulum  is  about  four  times  more  sensitive 
than  the  other  if  the  same  interference  method  is  used.  This,  however,  was 


not  desirable  at  the  outset,  the  above  method  of  reflection  admitting  of  easier 
interpretation.  Hence,  6=x/$D  nearly,  where  x  is  the  distance  apart  of  the 
two  slit  images  corresponding  to  the  distance  D. 

The  results  obtained  for  the  current  changes  of  inclination,  a,  in  seconds 
of  arc,  are  given  in  fig.  8,  on  the  same  plan  as  the  preceding.  The  work  was 
continued  for  nearly  two  weeks,  but  only  a  brief  example  is  shown  in  the 
figure. 

In  the  present  experiments  the  whole  weight  of  the  pendulum,  740  grams, 
rests  directly  on  the  lower  vertical  steel  pivot,  and  the  friction  is  correspond- 
ingly large  and  probably  excessive.  Static  friction,  however,  was  not  present, 
the  pendulum  being  continually  in  motion.  On  November  7  certain 
changes  of  apparatus  were  undertaken,  and  a  new  zero  had  to  be  selected. 
As  a  whole,  the  changes  of  inclination  of  this  pendulum  lie  within  1.5  seconds 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      11 

of  arc,  and  they  agree  in  order  of  values  with  the  data  of  the  steel  pendu- 
lum attached  to  the  pier.  Values  of  a,  however,  made  at  the  same  time, 
do  not  show  the  same  run  of  variation,  which  may  be  reasonable,  as  the 
present  pendulum  is  erected  on  the  concrete  subfloor  of  the  laboratory.  It 
was  supposed  that  some  change  of  inclination  would  result  from  the  pos- 
sible shifting  of  the  lower  pivot  in  its  socket  after  the  jar  following  an 
explosion,  but  there  is  no  certain  evidence  of  this.  The  chief  purpose 
of  these  experiments  is  thus  the  comparisons  which  they  will  offer  with 
the  cases  of  the  following  paragraphs  where  the  pendulum  is  partially 
supported  on  a  float. 

6.  New  apparatus,  with  float.  Horizontal  pivots.— A  thin  cylindrical 
sheet-copper  float  was  now  prepared,  about  10  cm.  in  diameter  and  10.3  cm. 
high,  weighing  with  appurtenances  175  grams.  It  was  attached  symmetri- 
cally to  the  axis,  and  the  method  of  submergence  is  shown  in  fig.  7,  where  K 
is  the  float  and  C  the  water-vat.  Three  vertical  brass  wires  about  2  mm.  in 
diameter,  120°  in  horizontal  angle  apart,  pass  from  the  disk  i  through  the 
surface  of  the  water.  The  buoyancy  was  found  to  be  627  grams  by  direct 
measurement,  whereas  the  remainder  of  the  pendulum  weighed  697.5  grams. 
The  effective  weight  was  thus  only  70.5  grams.  Subsequently,  to  increase 
the  moment  of  inertia  and  to  move  the  center  of  gravity  further  from  the  axis 
(in  the  interest  of  greater  permanence  at  the  pivots),  an  additional  weight 
was  attached  at  the  end  of  one  arm,  bringing  the  total  weight  to  M=p7i 
grams  and  the  effective  weight  M—  V  to  169.2  grams.  The  center  of  gravity 
of  the  solid  pendulum  was  moved  outward  from  8.1  cm.  to  13.1  cm.  from  the 
vertical  axis.  Moreover,  the  value  of  the  factor  is  now  M/(M— F)  =  5.74. 
Inasmuch  as  the  greater  part  of  the  weight  was  supported  by  the  float,  the 
pivots  were  here  tentatively  placed  horizontally,  the  lower  fitting  into  a  coni- 
cal hollow  of  glass-hard  steel  with  its  axis  horizontal;  but  the  results  shown 
below  are  quite  unfavorable  to  this  adjustment  of  pivot. 

To  find  the  moment  of  inertia  with  respect  to  an  axis  through  the  center 
of  gravity,  the  pendulum  was  swung  in  its  erect  position,  from  a  wire  of  known 
modulus  of  torsion.  In  this  way  the  moment  of  inertia  was  found  to  be 
i.6ioXio6  and  the  minimum  radius  of  gyration  4=40.71  cm.  Hence,  the 
square  of  the  effective  radius  of  gyration  was  i2=i2o+h=  1,830.  The  period, 
in  case  of  insignificant  damping,  was  determined  as  T=zo  seconds.  Finally, 
since  the  pendulum  is  supported  at  the  axis  and  not  at  the  center  of  gravity, 
the  new  or  flotation  center  of  gravity  is  thus  h'=Mh/(M—  V).  The  inclina- 
tion <p  of  the  axis,  or  v  —  wW/T^gh,  on  inserting  the  values  given,  reduces 
to  <p= 0.014  radian,  or  about  0.80  degree.  This  is  larger  than  necessary,  but 
it  was  thought  wise  not  to  diminish  it.  The  change  of  the  angle  of  inclina- 
tion, being  a  =  <p9,  for  the  deflection  0,  is  thus  determined.  For  a  given  <f> 
it  is  independent  of  the  presence  or  absence  of  the  float,  which  therefore 
does  not  conduce  to  enhance  the  precision  of  the  quantity  a,  except  in  dimin- 
ishing the  friction  of  the  pivots. 


12     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

The  force  at  the  distance  R  =  6 1.5  cm.  from  the  axis  (this  being  also  the 
position  of  the  line  of  light  passing  through  the  plate)  is,  for  like  6  and  <p,  since 


on  introducing  the  values  given, 


Thus,  if  AAT=  io~4  cm.,  FR=  io~3X  13  dynes,  or  about  io~3X4  dynes  per  van- 
ishing interference  ring. 

7.  Observations.  —  The  data  are  given  for  convenience  in  fig.  9,  on  the  same 
plan  as  the  above,  the  reflection  method  being  used.  Though  the  work  was 
continued  through  several  weeks,  only  an  example  is  shown,  as  it  was  neces- 
sary to  readjust  the  apparatus  frequently,  and  the  behavior  was  throughout 
anomalous.  Since  Q=x/^D, 

a  =  <f>0  =  0.0146  radian  =  0.81*  second  of  arc, 
since  D  =  9oocm.,  M  =971  grams,  M—  F=  169  grams, 

/?  =  6i.$  cm.,  &=i3.icm.       r=aosec. 

In  fig.  9  the  ordinate  x  is  given  in  arbitrary  units,  which  must  be  divided 
by  5.74  to  reduce  them  to  seconds  of  arc. 


ir 


The  conical  sockets  for  the  horizontal  pivots  of  the  pendulum  being  of  steel 
and  not  quite  smooth,  it  is  possible  that  the  relatively  enormous  values  of  the 
changes  of  inclination,  a,  registered  may  have  been  due  to  displacements  of 
the  pivots  in  their  sockets;  but  readjustments  of  the  fiducial  zero  (before  the 
observations  marked  n  in  the  graph)  were  as  frequently  necessary  when  there 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      13 


was  no  explosion  (marked  e  in  the  graph)  due  to  blasting  under  the  hill,  as 
after  the  occurrence  of  such  a  disturbance.  In  fact,  the  values  of  a  range 
within  26  seconds  and  are  often  as  large  as  that  per  day,  whereas  in  paragraph 
5,  for  the  case  of  the  apparatus  without  a  float,  the  whole  limit  of  variation 
was  not  above  1.5  seconds. 

In  fact,  the  two  slit  images  frequently  separated  to  a  distance  exceeding 
#  =  36  cm.,  or  nearly  as  many  seconds  (compared  with  the  isolated  maximum 
of  8  cm.  above),  while  the  distance  between  mirror  and  scale  was  but  .0  =  900 
cm.  (compared  with  2,000  cm.  above).  The  apparent  change  of  inclination 
of  the  line  determined  by  the  pendulum  pivots  is  thus,  in  the  present  case  of 
the  float,  registered  about  17  or  18  times  larger  than  the  above  similar  case 
without  a  float,  whereas  there  should  be  no  difference.  At  least,  it  is  difficult 
to  conceive  how  the  float,  which  practically  compensates  the  weight  of  the 
pendulum,  can  introduce  any  seriously  variable  torque  around  the  vertical  axis. 

In  fig.  ioa,  let  ABG  be  the  horizontal  projection  of  the  lower  pivot,  the 
center  of  buoyancy,  and  the 
center  of  gravity  of  the  pendu- 
lum. Then,  with  the  correspond- 
ing notation,  the  forces  involved 
will  be  A  +B  =  G,  the  couples 
involved,  Ah  and  Bh,  very  nearly, 
if  h  is  the  distance  AG.  The 
effective  couple  is  thus  Ch,  the 
vector  sum  of  Ah  and  Bh.  If 
the  angle  between  AG  and  BG  is 
e,  the  couple  Ch  may  be  resolved 
into  a  normal  couple  N  and  a 
parallel  couple  H  equal  to  Bhe 
nearly,  whose  axis  is  essentially 
horizontal.  In  fig.  1  06,  where  AZ  PIG.  10. 

is  the  vertical  and  AF  the  line  of 

pivots  at  an  angle  <p  to  AZ,  the  horizontal  couple  H  may  be  again  resolved 
into  a  couple  whose  axis  is  normal  to  AF,  which  is  ineffective,  and  another  P 
whose  axis  is  parallel  to  AF,  where  P  =  Bhe<p,  nearly.  Now,  although  P  is  of 
the  second  order  of  small  quantities,  it  does  not  follow  that  it  is  inappreciable, 
for  if  B  is  replaced  by  Vpg,  where  p  is  the  density  of  water  at  the  given 
temperature,  the  buoyancy  couple  is  Vpghep.  If  the  observed  deflection  is  B, 
the  couple  due  to  any  simultaneous  change  of  inclination  will  therefore  be 


and  the  tilt 

<x  =  <p(d 

Hence  only  if  p  remains  constant,  will 

af-a  =  <f>(6'-d) 
with  a  corresponding  value  free  from  c  for  T'—T. 


14     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


There  is  another  point  of  view  from  which  the  question  may  be  approached : 
Any  variation  of  buoyancy  B,  if  B  is  eccentric,  is  virtually  equivalent  to  a 
displacement  of  the  center  of  gravity  of  the  pendulum.  This  occurs  when 
the  temperature  of  the  water,  in  which  the  float  is  submerged,  changes.  Let 
AG  be  the  plane  through  the  axis  of  the  center  of  gravity  of  the  pendulum  when 
the  float  is  not  submerged,  k  the  perpendicular  distance  of  B  from  this  plane. 
The  center  of  gravity  after  the  submergence  of  the  float  will  be  displaced 
laterally  (if  Vp  is  the  mass  of  liquid  displaced  by  the  float) 

,f_Vpk_Yp 
~  M   ~  M** 

Since  the  center  of  gravity  must  lie  in  the  same  vertical  plane  with  the  line 
of  pivots  AF,  the  pendulum  will  have  to  rotate  over  an  angle 

e'  =  k'/h=VPs/M 

The  observed  angle  0  is  thus  to  be  divided  by  0'  to  obtain  the  amount  due  to 
simultaneous  changes  of  inclination  only.  Of  course,  e  may  be  either  positive 
or  negative.  Hence,  the  apparent  change  of  inclination  from  a  to  a'  is  to  be 
interpreted 


Before  discussing  the  question,  however,  it  is  preferable  to  obtain  data 
with  a  more  perfect  pivot  adjustment;  in  other  words,  to  use  pivots  inclined 
toward  the  center  of  gravity  and  provided  with  jeweled  bearings. 


c 


XI 


* 


FIG.  u. 


8.  Second  apparatus,  with  float.  Jeweled  bearings.—  The  anomalous 
results  for  a  obtained  in  the  last  experiments  were  in  the  first  place  to  be 
associated  with  the  unsatisfactory  pivots.  Hence,  these  were  readjusted  so 
as  to  point  toward  the  center  of  gravity  of  the  pendulum.  Moreover,  the 
steel  cup  was  inadequately  smooth  and  could  not  be  polished.  It  was  there- 
fore replaced  by  a  conical  hollow  of  polished  sapphire,  placed  so  that  its  axis 
prolonged  passed  through  the  center  of  gravity  of  the  horizontal  pendulum. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      15 

The  manner  in  which  the  jewels  were  secured  is  shown  in  fig.  na,  and 
sectionally  in  fig.  nb,  turned  at  right  angles  around  a  vertical  axis  to  the 
preceding  figure.  Here  a  is  the  lower  end  of  the  central  vertical  tube  of  the 
horizontal  pendulum  into  which  the  stem  b  of  the  forked  holder  c  fits  very 
snugly.  The  two  flat  prongs  c  carry  the  brass  screw  with  fine  thread  d,  which 
is  horizontal  and  is  secured  in  any  position  by  the  lock-nut  e.  The  conically 
hollowed  jewel,  black  in  figure,  is  firmly  embedded  in  the  small  brass  cylinder 
/,  which  in  turn  may  be  screwed  centrally  into  the  wider  cylinder  d  and  fixed 
by  a  lock-nut.  P  is  the  brass  stem  carrying  the  needle  of  glass-hard  steel 
which  dips  in  the  sapphire  socket.  The  cylinders  P  and  /  are  coaxial,  but 
may  be  given  any  inclination  to  the  vertical  and  then  locked.  As  the  effective 
weight  of  the  pendulum  does  not  exceed  170  grams,  the  strain  on  the  pin  and 
jewel  is  not  excessive,  and  the  results  appear  to  show  that  they  rendered 
excellent  service.  The  upper  pivot  played  in  a  groove  of  glass-hard  steel  as 
before,  and  it  did  not  seem  necessary  to  modify  this. 

The  remainder  of  the  horizontal  pendulum  was  of  the  form  already  sketched 
in  figs.  6  and  7 ;  but  precautions  were  subsequently  taken  to  mount  the  water- 
bath  for  the  float  on  a  separate  pillar,  quite  independent  of  the  horizontal 
pendulum  and  its  case.  Later  the  case  was  also  independently  mounted. 
Trial  was  made  of  a  water  damper  attached  to  the  end  of  the  beam  (H '  in 
fig.  6)  on  the  side  opposite  to  the  mirror  H.  This,  however,  was  soon  discarded 
because  of  the  capillary  forces  introduced.  As  a  rule,  the  damping  obtained 
at  the  float  is  adequate. 

In  order  to  set  the  zero  of  the  pendulum  at  a  given  point,  as  well  as  to  vary 
the  inclination  of  the  axis  by  the  definite  amount  needed  in  the  independent 
data  of  <p  (see  page  20),  the  lower  pivot  is  virtually  on  a  micrometer  screw, 
capable  of  moving  it  by  a  definite  amount  z  at  right  angles  to  the  plane  of 
the  pendulum.  This  device  is  shown  in  fig.  nc,  where  p  is  the  pivot  screw 
with  lock-nut  securing  it  to  the  brass  rod  /.  The  latter  fits  snugly  in  the  end 
of  the  piece  of  ^-inch  gas-pipe  k  and  is  secured  by  the  lock-nut  h,  the  pipe 
being  longitudinally  slotted  within  it.  The  rod  I  is  firmly  fastened  to  the 
micrometer  screw  m,  the  nut  of  which,  «,  drags  the  rod  /  from  left  to  right,  in 
spite  of  considerable  friction,  when  turned  clockwise.  Finally  the  rotation  of 
m  is  measured  on  a  dial  q  fixed  to  the  pipe  k.  Hence,  as  shown  below,  §  n, 
if  a  displacement  z  is  given  to  the  lower  pivot  at  a  distance  y  below  the 
upper,  a  =  z/y  =  <p0. 

9.  Observations. — The  constants  of  the  pendulum  were  the  same  as  in  the 
preceding  case,  with  the  exception  of  the  period  T,  which  was  found  to  be 
16.2  seconds.  The  other  constants  are  M  —  g^i  grams;  R  —  6i.s  cm.;  h=  13.1 
cm.;  M—  V=i6g  grams;  ^=40.7  cm.;  whence 

6  =  2 78 X io~*x;  a*=<pO  =  6X io-*x  radians,  nearly. 
To  obtain 


16     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


the  constant  T  only  had  to  be  changed,  the  former  value  of  <f>  being 
radian  and  T=  20  seconds.    Hence, 
/   400 


(I6.2)5 


=  0.0214  radian, 


or  about  1.2  degrees,  roughly.  This  is  an  unnecessarily  large  angle  and  is 
merely  admitted  as  a  first  experiment,  to  be  decreased  in  successive  experi- 
ments. Hence,  finally, 

a  =  <f>e  =  6.oXio-6x  radians  =1.23%  seconds  of  arc. 

The  apparatus  is  thus  relatively  insensitive,  seeing  that  1.2  seconds  go  to  a 
centimeter  of  deflection,  x. 


ZOO 


440 


fcO 
60 
40 
20 
0 

-20 
-40 
-60 
-80 


£5 


4 


9      // 

FIG.  12. 


60" 


ACT 


30* 


These  observations  were  carried  on  for  some  time  in  the  midst  of  other 
work,  during  February  and  March  of  1914,  the  method  of  reflection  being  used. 
The  data  are  first  given  in  fig.  1 2 .  The  ordinates  of  the  latter  are  in  arbitrary 
units  and  must  be  divided  by  5.74  to  reduce  them  to  seconds  of  arc.  In  the 
continuous  record  from  February  23  to  March  13,  the  range  of  variation  is 
almost  as  large  as  it  was  in  fig.  9,  showing  that  the  use  of  jeweled  bearings  has 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      17 


had  little  or  no  influence  on  the  result.  An  interesting  fact  is  the  depression 
produced  by  the  gale  on  March  i  (see  g  in  fig.  12).  Though  the  behavior  of  the 
apparatus,  as  such,  apart  from  the  anomalously  large  results,  was  throughout 
satisfactory,  it  was  supposed  that  the  attack  of  water  on  the  soldered  joints 
of  the  copper  float  was  an  objectionable  feature.  These  were  accordingly 
covered  with  resinous  cement,  with  a  removal  of  this  trouble  after  March  13, 
the  new  zero  being  indicated  at  n  in  fig.  12 ;  but  the  behavior  thereafter  was 
even  more  variable  than  before,  showing  that  something  not  connected  with 
the  change  of  inclination  is  in  question. 

10.  Observations,  continued. — The  horizontal  pendulum  was  now  read- 
justed for  greater  sensitiveness  and  for  a  smaller  vertical  inclination  <p,  by 
moving  the  upper  pivot  inward.  Since  T2  varies  as  i/<p,  considerable  displace- 
ment is  required  to  change  T  markedly.  The  period  found  was  T  =  2  5  seconds, 
all  the  other  constants  remaining  unchanged.  Hence,  since  the  original  period 
corresponded  to  7=  20  seconds,  ^'  =  0.014  radian,  the  inclination  is  now 

^  =  0.0147^  =0.0090  radian,  or  0.51°,  nearly. 
625 

Since  a  =  <pd  radians,  and  since  6=x/4D,  where  D  =  goo  cm., 

o.oooo         ,. 

a  =  — T-^—X  radians  =  0.5 1  <x  second. 
3,600 

Thus  a  centimeter  of  distance  between  the  two  slit  images  corresponds  to 
about  0.5  second  of  arc  of  inclination  of  the  pendulum  axis. 


/20 


A 


V 


\L 


10" 


c/' 


FIG.  13. 


The  damping  was  as  before  moderate,  due  only  to  the  axial  float. 

The  new  data  given  in  fig.  13  in  arbitrary  units  (to  be  reduced  5.74  times 
to  refer  them  to  seconds  of  arc)  show  the  same  peculiarities  as  the  preceding. 
The  range  of  variation  is  of  the  usual  abnormally  large  value.  There  is  no 


18     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

adequate  indication  of  increased  sensitiveness  due  to  larger  period  T  which 
is  in  question.  It  will  thus  be  necessary  to  endeavor  to  account  for  the  anoma- 
lous apparent  variability  of  a  observed  in  these  experiments  with  the  float. 

11.  Effect  of  temperature  on  the  float,  etc.— It  will  be  necessary  at  the 
outset  to  obtain  the  changes  of  buoyancy  due  to  corresponding  changes  of 
temperature  of  the  water  in  which  the  float  is  submerged.  To  obtain  the 
change  of  buoyancy  with  temperature,  the  following  table  of  increments  may 
be  consulted,  where  the  normal  temperature  is  taken  as  20°  and  V=&O2  cm.: 

t=io°  Vpt-VpK=  +1.179 

15  +  .706 

20  ±     .0 

25  -  .914 

30  -1-997 

The  total  effect  of  temperature  between  10°  and  30°  would  thus  be  but 
slightly  over  3  grams.  This  may  be  estimated  to  act  upon  a  lever  arm  not 
exceeding  i  cm.,  the  endeavor  having  been  made  to  keep  the  float  axial. 
Thus  we  may  assume  that  a  moment  of  3X981  dyne  cm.  would  not  be  ex- 
ceeded in  any  variation  of  temperature,  the  moment  being  0.15X981  per 
degree  centigrade. 

Direct  preliminary  experiments  on  the  effect  in  terms  of  the  deflection  x  of 
definite  moments  around  a  horizontal  axis  were  made  by  placing  10  gram 
weights  on  the  pan  (i,  fig.  7)  carrying  the  float,  at  a  distance  of  nearly  5  cm. 
on  either  side  of  the  vertical  axis  of  rotation.  The  effective  moment  is  thus 
95X981  dyne  cm.  The  successive  deflections  were  (differences  due  to  devia- 
tion of  pendulum  during  observation) 

10  grams  in  front,  £  =  3.4  cm.  10  grams  in  rear,  #  =  35. 4  cm. 

6-7  33-9 

This  is  equivalent  to  #  =  29.6  cm.  for  the  given  torque,  or 
29.6 


95X981 


0.31/981  cm. 


of  deflection  per  dyne-centimeter  of  torque.  Hence,  using  the  preceding  esti- 
mate, the  deflection  should  be 

0.31 

-g^-Xo- 15X981  =  0-46  cm. 

per  degree  centigrade  per  centimeter  of  eccentricity,  which  would  be  equivalent 
to  about  *=  i  cm.  for  a  range  of  temperature  from  10°  to  30°.  It  does  not 
appear,  therefore,  even  if  the  eccentricity  of  the  float  is  greater  than  was 
assumed,  that  the  temperature  decrement  can  be  a  menace. 

The  endeavors  to  measure  the  temperature  discrepancy  directly  were  all 
failures,  inasmuch  as,  during  the  long  time  of  cooling,  the  deviations  of  the 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      19 

pendulum  exceeded  the  temperature  effect,  and  because  the  necessary  stirring 
of  the  water  in  the  float  interfered  with  the  free  play  of  the  pendulum.  The 
temperature  effects  obtained  were  as  liable  to  be  positive  as  negative.  In 
fact,  it  is  conceivable  that  although  direct  effect  of  temperature  may  not  be 
serious,  the  indirect  effect  produced  by  the  friction  of  irregular  convection 
currents  of  water  on  the  float  may  be  so.  Unfortunately  no  means  of  allowing 
for  these  suggests  itself,  so  that  constancy  of  temperature  is  a  condition  for 
the  proper  functioning  of  the  floated  pendulum.  Symmetrical  occurrences 
would  of  course  be  ineffective. 

It  will  now  be  advisable  to  resume  the  equation  of  moments  in  §  7,  where 
the  torque  is  fully  expressed  as 


if  the  weight  mg  is  put  on  the  pan  of  the  float  at  a  distance  /  from  the  vertical 
axis,  T—  T=mgl<p.  Consequently  if  p  does  not  vary,  the  effect  of  e  vanishes, 
whence 

,  =       ml 

M(e'-ey 

the  distance  of  the  center  of  gravity  from  the  axis,  may  be  found.  The  meas- 
urements below  show  this  to  be  a  good  method  under  proper  precautions. 

A  number  of  experiments  of  this  kind  were  made  with  the  pendulum  mod- 
ified by  the  addition  of  a  damper  on  the  left,  which  would  throw  the  center  of 
gravity  slightly  in  that  direction.  The  first  two  series  of  observations  com- 
pared the  deflection  x  with  the  water  damper  attached  and  the  float  either 
fully  or  less  than  half  submerged,  respectively.  The  variable  data  obtained 
for  x  and  their  small  value  showed  that  capillary  forces  were  in  play  which 
completely  vitiated  the  use  of  the  pendulum.  The  zero  was  not  steady. 
There  seemed  to  be  an  actual  capillary  resistance  in  play.  Hence  the  h 
obtained  was  too  large.  The  damper  at  the  end  of  the  arm,  notwithstanding 
its  convenience,  is  therefore  not  admissible  as  an  attachment  under  the 
conditions  of  sensitiveness  of  the  pendulum. 

With  the  damper  removed,  the  float  being  but  half  submerged,  the  results 
were  improved,  but  the  capillary  forces  on  the  float  were  still  excessive.  It  was 
not  until  the  float  was  completely  submerged  that  the  capillary  forces  were 
negligible  and  consistent  values  of  h  were  obtained,  the  accuracy  of  which 
might  easily  have  been  improved  by  closer  observation. 

The  addition  of  over  500  grams  weight  of  buoyancy  has  no  effect  on  the 
deflection,  if  the  error  due  to  capillary  forces  is  allowed  for. 

If  the  torque  applied  is  constant  while  the  temperature  of  the  water  in  which 
the  float  is  submerged  changes,  the  differential  equation  becomes  dQ/dp  = 
tV/M  or 


20     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


Thus  the  compensation  would  be,  per  degree  of  e,  at 

10°  io6dp=  -J-I47  io9dx=   76 

IS  +  88  46 

20  ±o  o 

25  -  94  -  49 

30  -259  ~I29 

Hence,  for  the  whole  range  of  20  degrees,  the  compensation  would  not  exceed 
#  =  0.2' per  degree  of  e,  or  about  x=  i  cm.  for  e=  5°.  This  estimate  is  thus  of 
the  same  order  as  the  above,  since  the  eccentricity  of  i  cm.  there  postulated 
makes  e=X*  radian  or  5  degrees,  roughly. 

Finally,  inasmuch  as  considerable  alteration  had  been  made  at  the  pendu- 
lum during  the  preceding  experiments  (addition  of  damper,  etc.),  it  seemed  ad- 
visable to  redetermine  <f>,  using,  however,  the  micrometer  method,  instead  of 
the  more  elaborate  pendulum  method.  The  following  values  of  x  were 
found  for  successive  turns  of  6°  each  of  the  micrometer  screw: 


Turn  of  screw. 

X 

Mean  Sx  per  6°. 

0° 

29.0  cm. 

18.5  cm. 

6° 

11.9 

12° 

-  7-5 

18° 

-25-7 

24° 

-44-5 

The  screw  being  a  K'-inch  screw  with  20  threads  to  the  inch,  its  pitch  may  be 
put  0.125  cm.  If  z  is  the  displacement  of  the  lower  pivot  for  each  partial  turn 
of  6°,  y  the  distance  apart  of  the  pivots,  and  C  the  constant  to  be  found, 

a  =  z/y—Cx  radians,  or,  in  seconds  of  arc,  C— 0-31 

The  value  of  C  found  for  the  pendulum  used  in  the  case  above  was  C= 
0.515.  The  difference  is  larger  than  was  expected;  but  with  the  center  of 
gravity  but  12  cm.  from  the  axis,  the  addition  or  removal  of  the  weights  at 
the  end  of  the  beam  60  or  70  cm.  from  the  axis  is  of  marked  consequence.  It 
is  also  surprising  that  the  displacement  method  is  so  consistent  in  its  results, 
as  nothing  more  than  an  ordinary  clock-dial  with  a  pointer  was  used  at  the 
micrometer.  These  results  could  easily  be  much  improved.  In  other  words, 
the  present  direct  displacement  method  for  C=<p/^D,  and  the  above  direct 
momental  method  for  h,  are  not  only  much  more  simple,  but  on  the  whole 
more  reliable  than  the  laborious  vibration  methods.  With  M  they  suffice 
completely  for  the  evaluation  of  the  torque,  T=Mgh<pe. 

12.  Further  observations. — A  final  series  of  observations  was  now  made 
with  the  new  apparatus,  recording  under  good  conditions,  in  the  absence  of 
artificial  heating.  The  trough  for  the  float,  supported  entirely  free  from  the 
horizontal  pendulum,  was  provided  with  a  thermometer,  which  was  read  off 
at  the  same  time  as  the  inclination  observation.  The  direct  measurement 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      21 


of  <f>  gave  a  =  o.^x  seconds  of  arc.  These  (summer)  data  are  given  in  fig.  14 
and  the  temperatures  are  inserted  in  the  same  figure.  The  work  was  continued 
for  about  6  weeks,  not  all  of  the  data  finding  room  in  the  figure,  and  the  graph 
after  July  3  had  to  be  displaced,  as  shown. 

The  new  results  still  partake  of  the  same  tendency  to  enormous  variations 
which  characterize  the  older  (winter)  data.  The  essential  error  has,  therefore, 
not  been  removed.  On  comparison  with  the  detailed  temperature  curve  above, 
however,  the  clue  of  the  anomaly  is  obtained,  for  although  the  temperature 
variations  are  not  quite  contemporaneous  with  those  of  inclination,  there  can 
be  no  doubt  of  the  immediate  relation  between  them.  The  case  is  all  the  more 


cfcnp. 


'\ 


AV 


w 


-J 


FIG.  14. 

puzzling,  however,  as  single  degrees  are  in  question,  enormous  changes  of 
inclination  being  produced  by  4  degrees.  Under  the  circumstances,  moreover, 
complete  identity  in  the  direction  of  variation  of  temperature  and  inclination 
graphs  was  not  to  be  expected,  for  the  temperatures  given  are  those  of  the 
water  in  the  float  and  will  therefore  vary  more  sluggishly  than  the  tempera- 
ture of  the  metal  parts.  The  air  temperatures,  again,  which  were  also  taken, 
would  vary  faster  than  those  of  the  metal,  evidence  for  which  will  presently 
be  shown.  It  is  therefore  next  in  order  to  actually  examine  the  structure  of 
the  standard  of  the  horizontal  pendulum. 

13.  Effect  of  temperature  on  the  scaffolding.— To  give  the  columnar  sup- 
port of  the  horizontal  pendulum  adequate  steadiness,  it  was  braced  from  be- 
hind as  shown  in  fig.  15.  It  was  not  foreseen  that  any  menace  could  lurk  in 
such  a  system,  such  as  was  later  detected.  In  fig.  15,  ABC  is  a  side-view  of 
the  brass  vertical  standard  (in  duplicate,  as  shown  in  fig.  7),  the  horizontal 


22     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

pendulum  being  supported  between  A  and  B,  while  the  heavy  base  CD  rests 
on  foot-screws  on  the  cement  subfloor  of  the  basement  of  the  laboratory. 
BD  is  the  brace  in  question,  extending  almost  half-way  up  the  standard  at 
an  angle  of  about  0=14°  to  it.  For  ordinary  quiet  surroundings  this  truss 
seemed  to  be  adequate,  as  the  water-vat  of  the  float  was  held  on  a  separate 
standard,  free  from  the  pendulum  and  its  case.  The  pendulum,  in  view  of 


FIG.  17. 


the  float,  was  therefore  virtually  very  light.  The  difficulty  encountered  resides 
in  the  fact  that  even  small  differences  in  the  coefficient  of  expansion  of  BC 
and  BD  will  seriously  tilt  the  axis  AC.  To  express  these  relations  let  h,  v,  b, 
be  the  hypothenuse,  the  vertical,  and  the  base  of  a  right-angled  triangle  as 
shown  in  fig.  15  and  idealized  in  fig.  16.  Let  </>=dh/h=db/b  for  the  same 
temperature  increment  of  i  °  C.  be  the  coefficient  of  expansion  of  the  base  and 
of  the  brace  (for  convenience),  and  fi=dv/v  that  of  the  brass  post.  Then  it 
follows  easily  that  for  an  increase  of  i  °  C.  of  the  environment, 

fe¥-l#4W-&0((<H-0)  cos  a+sin  ado) 

where  a  is  the  angle  of  inclination  of  the  post  and  da  its  increment.  Hence, 
since  a  =  90°,  very  nearly, 

da  =  (^-/9)  /tan  0 
Since  tan  B  =  0.2  5,  nearly, 

da  =  $(4'—  0)  radians 

or,  for  the  above  a  =0.3*  seconds, 


Hence,  so  small  a  difference  of  coefficient  of  expansion  as  4>-p=io~*  would 
give  rise  to  a  deflection  of  dx  =  2.7,  nearly  3  cm.  per  degree  of  increase  of  tem- 
perature. In  fact,  this  arrangement  actually  suggests  itself  as  a  remarkably 
sensitive  method  for  measuring  small  elongations;  for,  since  da=  (<}>  —  &)  cot  B, 
independent  of  all  lengths,  da  increases  as  0  decreases  without  limit,  and  the 
question  is  merely  one  of  experimental  adjustment.  If  the  hypothenuse  h 
alone  expands,  the  remaining  temperatures  being  kept  constant, 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      23 
or  for  the  above  data 


quantities  of  the  same  order  as  the  above.  Thus  if  dx  =  i  cm.  (for  the  rela- 
tively small  distance  of  scale  from  mirror,  D  =  goo  cm.),  0  =  o.3sXio6,  and  it 
should  be  easy  to  measure  one-tenth  of  this  expansion. 

The  peculiar  interest  which  attaches  to  this  equation  for  da  or  any  corre- 
sponding case  is  the  absence  of  all  need  of  length  measurement  in  the  combina- 
tion. In  the  right-angled  triangle  hvb,  fig.  15,  it  is  merely  the  angle  6  which 
must  be  given,  all  the  quantities  compared  being  numbers.  Of  course,  the 
relation  of  x  and  a  remains,  into  which  the  distance  of  the  mirror  from  the 
scale  will  enter.  In  the  complete  equation  (if  da  is  replaced  by  a) 


<p  may  be  found  directly  as  shown  above. 
If  the  interferometer  is  used,  x/q.D  is  to  be  replaced  by  AAT/2.R  so  that 


If  values  of  the  above  order  be  inserted,  i.e.,  <p  =  o.oi  ;  AAf=  io~4;  R  =  io2;  tan 
6  =  0.2  5  ;  then  0  —  /3  =  1  2  X  io~10.  In  case  of  the  other  equation,  da  =  a^/sin  20, 

^=12X10-'° 

In  any  case,  therefore,  an  expansion  of  the  order  of  4Xio~10  per  vanishing 
interference  ring  (AAf=3/io5)  should  be  measurable.  This  seems  by  far  the 
most  sensitive  arrangement  for  measuring  elongations  which  has  yet  been 
proposed.  The  full  equation  in  question  would  be 


from  which  any  of  the  above  forms  follow  at  once. 

In  its  bearing  on  the  horizontal  pendulum,  the  above  result  is  fatal.  Braces 
of  all  types  will  have  to  be  discarded.  The  following  incidental  experiments 
will  bear  this  out:  The  brace  was  heated  with  a  single  rapid  brush  of  the 
Bunsen  flame,  such  as  would  not  have  imparted  any  easily  appreciable  in- 
crease of  temperature  to  the  massive  rod.  The  times  of  observation  were 
also  recorded,  the  results  being  as  follows : 


Time. 

xcm. 

Remarks. 

9h28m  
30  
40  

2.2 
6.0 
5.O 

Zero  position. 
Quick  brush  of  brace  with  Bunsen  flame. 
Cooling. 

4S 

4.c 

Do. 

Qh  A5m 

20  8 

Additional  brush  of  brace  with  flame. 

55    

23.7 

Cooling. 

io  15  

14.0 

Do. 

26 

11.4 

Do. 

42 

Q.S 

Zero  at  2.2  cm. 

24     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

These  results  are  quite  regular.  The  extremely  slow  cooling  shows  how 
near  the  temperature  is  to  that  of  the  environment,  the  temperature  excess 
being  nearly  negligible.  It  shows  also  how  difficult  it  is  to  obtain  rigorous 
temperature  constancy  in  the  metallic  truss  exposed  to  the  surrounding  at- 
mosphere. Thus  it  would  have  taken  considerably  over  two  hours  to  dissipate 
the  negligible  difference  of  temperature  in  the  last  case. 

Again,  a  Bunsen  flame  placed  about  30  cm.  from  the  brace,  about  at  e  in 
fig.  15,  gave  the  following  result: 

9h  i5m        *  =1.5  cm.     Cold.    Burner  placed  as  stated. 

20  3.0  Gradually  heating.    Burner  now  removed. 

28  2.2  Cooling. 

The  radiation  of  the  Bunsen  burner  at  a  distance  of  about  i  foot  is  thus 
quite  perceptible,  in  spite  of  the  fact  that  the  brace  and  standard  are  to  some 
extent  affected  differentially. 

With  this  experiment,  therefore,  the  mysterious  temperature  variation  has 
been  cleared  away,  and  it  provides  definite  specifications  for  the  installation 
of  such  an  apparatus.  I  will  only  add  that  similar  experiments  tried  on  the 
scaffolding  of  the  permanent  mirrors  produced  only  negligible  effects.  Thus 
a  Bunsen  flame  run  rapidly  along  any  of  the  horizontal  braces  changes  the 
deflection  x  only  a  few  millimeters. 

14.  Inferences. — If  we  abstract  from  discrepancies  introduced  by  the  pendu- 
lum truss,  which  are  to  be  separately  treated,  it  may  be  assumed  that  the  data 
obtained  with  the  partially  floating  pendulum  represent  the  actual  tilting  of 
the  concreted  subfloor  of  the  laboratory.  With  a  reasonably  constant  tem- 
perature in  a  cellar  room,  in  the  absence  of  artificial  heat,  temperature  dis- 
crepancies should  no  longer  be  seriously  menacing.  To  account  for  the  differ- 
ence between  the  small  variations  of  a  obtained  in  the  absence  of  a  float  and 
the  large  variations  on  addition  of  the  float  to  the  same  pendulum,  it  is  suffi- 
cient to  admit  that  the  friction  at  the  vertical  pivots  in  the  former  case  (float 
absent)  was  excessive,  and  that  the  full  deflection  can  not  appear,  unless  the 
weight  is  taken  off  the  pivots  as  in  the  latter  case  (float  present).  This  is 
particularly  the  case,  since  the  inclination  resulting  from  the  expanding  brace 
presently  to  be  mentioned  should  in  any  case  have  been  present.  Capillary 
forces  at  the  float,  mounted  axially  as  above,  have  produced  no  appreciable 
distortion,  as  they  did  when  the  water  damper  was  mounted  at  the  end  of  the 
beam.  Finally,  the  float  is  itself  a  sufficient  damper,  and  in  the  absence  of 
air-currents  the  front  of  the  case  may  actually  be  kept  open,  as  was  done  in 
most  of  the  later  experiments,  in  a  room  free  from  artificial  heat.  Moreover, 
it  does  not  seem  necessary  to  construct  the  floating  horizontal  pendulum  on  so 
large  a  model  as  was  done  in  the  above  paper,  so  that  a  smaller  portable  model 
may  be  a  serviceable  instrument  for  many  laboratory  purposes,  seeing  that  the 
constants  are  determinable  by  the  direct  method  indicated.  In  how  far  the 
sensitiveness  may  be  increased  by  applying  the  buoyant  force  at  the  center 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     25 

of  gravity  or  elsewhere  can  not  be  answered,  as  it  is  not  unlikely  that  the 
capillary  forces  introduced  will  in  such  a  case  be  a  serious  consideration. 

The  most  interesting  result  obtained  is  the  effect  of  temperature  on  the 
inclined  brace  supporting  the  vertical  standard  of  the  horizontal  pendulum. 
It  appears  that  even  insignificant  differences  in  the  coefficient  of  expansion  of 
these  parts  are  at  once  manifested  as  an  appreciable  change  of  inclination 
varying  with  temperature.  In  fig.  14,  for  instance,  if  the  large  oscillations  be 
interpreted  as  a  temperature  effect,  one  may  estimate  that  the  change  of  i  °  C. 
of  the  temperature  of  the  environment  is  equivalent  to  a  deflection  of  x= 8  cm. 
at  the  scale,  or  equivalent  to  about  a  =  2. 5  seconds  of  change  of  inclination. 
It  is  therefore  necessary  to  avoid  this  deficiency  of  the  apparatus  with  scrupu- 
lous care ;  in  other  words,  to  avoid  all  lateral  bracing  if  the  material  can  not 
be  guaranteed  as  rigorously  homogeneous.  Hence,  in  the  more  refined  experi- 
ments, the  pendulum  is  to  be  swung  with  advantage  from  a  single  sufficiently 
stiff  metallic  post  anchored  in  the  ground.  Moreover,  since  the  pendulum 
with  a  trough  for  the  float  supported  quite  free  from  the  pendulum  is  virtually 
very  light,  a  standard  made  of  a  length  of  i-inch  gas-pipe  well  anchored  in  the 
ground  seems  to  be  most  promising.  The  case  and  the  optical  apparatus  are 
in  every  instance  to  be  supported  entirely  free  from  the  horizontal  pendulum. 

The  errors  which  have  been  detected  in  the  case  of  the  braced  pendulum 
are  in  all  probability  also  present  in  the  case  of  a  pier,  if  the  pier  confronts 
the  illumination  of  the  room  or  the  heating-pipes  of  the  building  on  one  side 
only.  In  such  a  case,  the  exposed  side  will  expand  on  the  cold  side  as  an  axis, 
and  a  tilting  of  the  pivotal  line  of  the  horizontal  pendulum  must  result.  Un- 
fortunately, this  is  the  condition  to  which  the  large  pier  in  our  laboratory  is 
subject  and  which  it  is  impossible  to  remedy.  It  is  probable  that  the  excur- 
sions observed  with  the  steel  pendulum  in  §  3  are  largely  to  be  interpreted  in 
this  way.  Temperature  observations  will  here  be  of  little  avail,  since  then- 
distribution  in  the  immense  mass  of  masonry  is  in  question.  Similarly,  the 
absorption  and  release  of  moisture  when  an  unavoidably  heated  basement 
room  passes  from  the  damp  summer  to  the  dry  winter  conditions  may  have 
a  similar  tilting  effect. 

15.  The  precision  measurement  of  elongations. — From  another  point  of 
view  the  exceedingly  sensitive  expansion  apparatus  which  has  been  described 
is  interesting  on  its  own  account.  In  the  diagram,  fig.  16,  the  hvb  triangle 
supports  the  horizontal  pendulum  PP,  normal  to  its  plane,  on  the  pivotal 
hangers  pp;  g  (outward  from  the  plane  of  the  figure)  is  the  grating  at  the  end 
of  the  pendulum,  n  and  m  the  concave  mirrors  of  the  displacement  interfer- 
ometer. Apart  from  the  instrumental  (elasticity  and  viscosity,  etc.)  and  en- 
vironmental conditions,  such  an  apparatus  should  register  expansions  dl/l  of 
an  order  even  smaller  than  4Xio~10  per  vanishing  interference  ring,  for  the 
registered  sensitiveness  of  the  above  apparatus  could  easily  be  increased.  No 
doubt  much  of  this  would  be  taken  up  by  the  yield  of  the  apparatus;  but 
nevertheless  it  is  over  io4  times  smaller  than  the  expansion  of  an  average 


26     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

metal  per  degree  centigrade.  The  actual  limit  is  necessarily  an  experimental 
question. 

As  an  example  it  may  be  worth  while  to  determine  whether,  potentially, 
the  apparent  contractions  of  rods  lying  longitudinally  in  the  direction  of  the 
earth's  motion  could  actually  be  measured  and  to  what  degree.  With  this 
end  in  view  the  problem  may  be  stated  with  reference  to  fig.  17,  where  P'P' 
is  the  polar,  EE  the  equatorial  diameter  of  the  earth,  p,  p'  two  diameters  in 
latitude  23.5°.  The  motion  of  the  earth  takes  place  along  the  diameter  p' 
with  a  mean  speed  3  X  iQ-6  cm./sec.  The  hvb  triangle  of  fig.  16  is  set  up  with 
the  side  v  vertical  in  latitude  23.5°  and  the  base  horizontal  and  in  the  plane  of 
the  meridian  as  shown.  The  side  v  carries  the  horizontal  pendulum  P  with 
its  plane  normal  to  the  meridian,  the  line  of  observation  being  mn  in  the  merid- 
ian. The  excursions  of  the  grating  on  the  pendulum  are  read  off  on  a  linear 
displacement  interferometer,  the  framework  and  the  two  component  beams 
running  in  the  same  direction.  All  parts  of  the  instrument  are  therefore 
identically  subject  to  the  same  effects. 

Twelve  hours  of  rotation  place  the  triangle  in  the  opposed  position  hW, 
and  the  question  to  be  determined  is  the  change  of  angle  a  (nearly  90°)  result- 
ing, seeing  that  the  relation  of  all  the  sides  to  the  motion  of  the  earth  has  been 
changed  and  v'  instead  of  h  moves  parallel  to  it.  For  convenience  in  computa- 
tion, the  angle  0  may  be  roughly  taken  as  45°  instead  of  47°,  so  that  h*  =  v*+b* 
=  2v2.  In  this  case  we  may  write 

(i)  da  =  2dh/h  -  dv/v  -  db/b 

by  reducing  the  equation  for  da  above.  It  should  be  noticed  that  the  equation 
is  purely  numerical,  the  degree  being  zero. 

Let  v'  be  the  velocity  of  the  earth,  c  the  velocity  of  light,  so  that  @  =  v'/c  = 
io~4  and  Vi-/32  is  the  longitudinal  contraction  coefficient.  The  size  of  the 
parts  hvb  and  h'v'b'  under  conditions  of  motion  may  therefore  be  replaced 
respectively  by 

\/2fl 


whence,  nearly, 


4        44        2 

On  the  displacement  interferometer 


(3)  da  =  <p 

where  *  is  the  inclination  of  the  axis  of  the  horizontal  pendulum  in  radians, 
AAT  the  micrometer  displacement  at  one  of  the  opaque  mirrors  corresponding 
to  the  two  positions,  R  the  distance  of  the  grating  at  the  end  of  the  horizontal 
pendulum  from  its  axis.  Incorporating  equation  (3)  finally, 


which  is  the  required  equation. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      27 

In  the  apparatus  used  above,  <p=io~z  and  R=io*  cm.,  /32=io-8.  These 
are  moderate;  the  former  could  easily  be  reduced.  Hence, 

AAr     io-*Xio8 

—  icr*  —  =          Cm' 

Even  in  the  case  of  the  present  apparatus,  therefore,  about  three  interference 
rings  should  vanish  (potentially)  between  the  positions  hvb  and  h'v'b'. 

A  similar  comparison  might  be  made  for  the  position  of  the  horizontal 
pendulum  normal  to  the  plane  of  the  diagram,  in  relation  to  the  first  and  final 
conditions  discussed.  But  the  lines  are  now  oblique  and  require  two  or  more 
projections,  and  this  additional  complication  is  superfluous  here. 

Contractions  of  the  pendulum  itself  must  be  negligible,  as  these  merely  dis- 
place the  center  of  gravity  in  the  plane  of  the  pendulum  and  are  otherwise 
not  amplified.  Tidal  forces  have  approximately  the  same  value  in  the  two 
positions,  or  may  be  allowed  for.  There  remains,  therefore,  the  contraction 
of  the  earth  itself,  which  changes  from  a  sphere  of  radius  r  to  an  oblate  ellip- 
soid with  its  minor  axis  r  \/i  —  /S2  in  the  direction  of  motion  pf.  In  a  general 
way  we  may  state  at  the  outset  that  as  the  triangle  is  a  part  of  the  earth,  its 
distortion  could  not  be  recognized  for  the  lack  of  an  independent  base  of  com- 
parison. But  the  question  is  advantageously  approached,  specifically,  as  fol- 
lows. Fig.  17,  which  contains  the  sphere  and  the  ellipsoid  in  question,  shows 
that  the  diameter  p  is  displaced  to  q  and  that  the  angle  a  moves  over  a  distance 

s=rdO/sin  6  nearly 

where  dd  is  the  angle  between  p  and  q  and  0  =  45°  nearly.  But  the  displace- 
ment 5  is  the  contraction  of  r  cos  0  or 


Hence, 


The  same  angular  deviation  occurs  between  the  two  tangents  or  bases  pro- 
longed, since  the  equation  of  the  ellipse  referred  to  the  circumscribed  circle  is 


fiP 

dd  =  —  sin  20  =  0*/4  nearly. 

Hence,  the  two  angles,  if 

da  =  2dd=  fP/2  as  above 


o^mfi  that  in  the  position  h'v'b'  the  angle  a  does  not  change.    The  displace- 
ment from  p  to  q,  therefore,  keeps  the  center  of  gravity  in  the  normal  plane  in 


28     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

which  these  lines  are  traces  and  no  effect  could  be  recognized.  Similarly  if  the 
horizontal  pendulum  were  attached  to  a  large  massive  vertical  pendulum 
(rigid  plumb-line)  the  displacement  da/2  would  escape  detection.  Neverthe- 
less the  potential  possibility  of  the  method,  well  illustrated  by  this  example, 
seemed  to  make  it  worth  while  to  endeavor  to  develop  it,  for  there  are  other 
non-compensated  micrometric  deviations  of  the  earth's  diameter,  to  which  it 
would  be  directly  applicable. 

16.  Improved  pendulum.  —  The  suspension  of  the  symmetrical  pendulum 
was  now  modified  so  as  to  embody  the  suggestions  contained  in  the  above 
work.    The  two  pivots  were  supported  on  a  single  post  of  i-inch  gas-pipe, 
sunk  into  a  hole  in  the  concreted  subfloor  of  the  basement  and  secured  with 
plaster  of  paris.    It  was  hoped,  in  this  way,  to  obviate  the  possibility  of  tem- 
perature disturbances  in  their  immediate  effect  on  the  pendulum.    Naturally, 
the  post  was  insulated  from  every  other  part  of  the  apparatus,  so  that  the 
pendulum  was  quite  free  and  independent.    Its  tin  case  was  adjustably  sup- 
ported on  the  iron  scaffolding  carrying  the  mirrors,  while  the  tank  for  the  float 
rested  on  an  independent  standard  rising  from  the  subfloor  in  question.    It  is 
improbable  that  short  of  a  special  brick  pier  for  the  instrument  a  more  advan- 
tageous method  of  mounting  could  have  been  devised.    It  was  therefore  in- 
teresting to  observe  how  the  whole  apparatus  would  behave,  on  transition 
from  fall  to  winter  conditions;  i.e.,  to  find  the  effect  produced  on  turning  on 
the  steam  heat  of  the  laboratory.    The  observations  are  given  in  the  next 
paragraph. 

17.  Observations  with  the  new  pendulum.  —  These  observations  are  given 
in  fig.  1  8  in  the  usual  way,  the  inclinations  a,  in  seconds  of  arc,  being  con- 
structed in  their  variations  with  time.    To  determine  the  constants  of  the 
apparatus  the  micrometer  method,  similar  to  the  above  (§  n,  end),  was  em- 
ployed.    The  variation  of  x  due  to  a  twist  of  12°  of  micrometer  screw  was 
found  to  be  £  =  27,  26,  26  cm.,  or  on  the  average  2.2  cm.  per  degree,  whereas 
the  former  value  was  #  =  3.1  cm.    Hence,  since  the  other  constants  are  the 
same  as  above  (the  distance  apart  of  the  pivots  being  75  cm.  and  the  pitch  of 
the  screw  0.125  cm.,  or  3.  47X10*  cm.  per  degree  of  arc) 


or  a  =  0.43*  second  of  arc,  nearly.  This  constant  was  used  in  the  reductions. 
In  addition  to  the  deviations  x,  the  temperature  of  the  room  and  the  weather 
conditions  outside  were  taken  daily.  The  latter  showed  no  consistent  influ- 
ence and  will  be  disregarded  here. 

The  first  branch  of  the  a.  curve  (A  A),  from  July  16  to  August  23,  shows  an 
initial  ascent  until  July  20  and  thereafter  a  fairly  uniform  descent.  The  tem- 
peratures (as  shown  by  the  temperature  curve,  fig.  18)  during  the  first  half 
of  the  observation  period  might  suggest  some  relation,  but  they  quite  fail 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      29 

to  do  this  during  the  second  half,  where  temperature  in  general  rises  and  the 
a  curve  falls.  The  curves  may  be  real  and  indicate  a  gradual  settling  of  the 
ground  during  the  whole  period,  modified  by  rains,  etc. ;  or  there  may  have 
been  a  gradual  viscous  yield  of  the  support  of  the  pendulum  in  its  concrete  base. 
Whatever  be  its  nature,  the  pendulum  fully  recovers  from  this  apparently 
continuous  subsidence  during  the  second  period  of  observation  (BB),  between 
August  23  and  September  28,  omitting  the  gaps  at  a  and  b.  The  temperature 
observations  (not  drawn)  show  no  relation  to  the  a  curve  whatever.  The 
curve  being  undulatory,  it  can  not  be  referred  to  any  persistent  yielding  or 


other  similar  discrepancy.    It  is  probable,  in  fact,  that  both  curves  AA  and 
BB  show  the  actual  tilting  of  the  concreted  subfloor  of  the  laboratory. 

On  September  28  the  steam  heat  was  turned  on  (scale  of  a,  on  the  right)  and 
the  totally  new  character  of  both  the  a  curve  CC  and  the  temperature  curve 
furnish  abundant  evidence  of  the  importance  of  this  disturbance.  In  fact,  the 
pendulum  behaves  at  first  like  an  extremely  sensitive  thermometer.  Observing 
that  the  temperature  scale  is  enormously  smaller,  rise  and  fall  of  temperature, 
i.e.,  depression  and  elevation  of  both  curves,  may  in  general  be  coordinated 
throughout.  But  there  is  no  quantitative  relation  between  the  two  curves. 
Thus  the  marked  rise  of  temperature  from  about  16°  to  nearly  30°  at  the  end 


30     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

of  September  and  beginning  of  October  shows  an  a  effect  in  the  curve  following 
c  quite  inferior  to  the  effect  later  between  g  and  h,  when  the  changes  of  tem- 
perature are  much  less  marked.  Moreover,  the  minimum  between  /  and  g  is 
much  too  large  to  be  associated  with  the  mean  temperature  minimum,  and  in 
fact  they  do  not  coincide.  The  determined  rise  of  the  curve  at  g  began  much 
before  any  corresponding  temperature  change.  The  gaps  at  d  and  e  introduce 
uncertainties,  but  nevertheless  one  would  have  expected  an  a  minimum  there. 
It  seems  probable,  therefore,  that  temperature  does  not  act  directly  on  the  pen- 
dulum (expansion  of  its  parts)  but  acts  on  it  through  another  system,  which 
is  probably  the  house  itself.  The  marked  effect  of  steam  heat  is  to  thoroughly 
dry  out  the  basement  room.  It  may  be  inferred  that  this  is  accompanied  by 
redistributions  of  the  stresses  of  the  building  and  that  the  concrete  subfloor 
responds  to  the  alteration  of  load.  It  should  be  noted  that  the  last  curve  CC 
has  been  dropped  7  seconds  to  accommodate  it  in  the  drawing  and  that  there- 
fore the  a  curve  CC  as  a  whole  lies  much  above  the  original  a  curve  A  A, 
although  the  average  temperatures  are  not  very  different.  It  seems  improb- 
able, therefore,  since  the  curve  has  much  more  than  recovered,  in  fact  has 
considerably  exceeded  its  original  reading,  at  about  the  same  temperature, 
that  there  can  here  be  any  viscous  yielding  in  the  apparatus  itself.  Further 
consideration  will  be  given  in  the  next  section  in  connection  with  the  steel 
pendulum. 

Finally,  since  the  inevitable  variations  here  recorded  are  within  16  seconds 
of  arc,  it  was  out  of  the  question  to  attempt  to  attach  the  interferometer  appa- 
ratus, adapted  for  reading  within  hundredths  of  a  second.  The  work  was  there- 
fore abandoned. 

PART  II.— AN  APPLICATION  OF  THE    DISPLACEMENT  INTERFEROMETER 
TO  THE  HORIZONTAL  PENDULUM. 

18.  Introductory. — The  displacement  of  ellipses  or  of  interference  lines  in 
the  spectrum  is  probably  capable  of  being  photographed  for  continuous  reg- 
istry, though  less  easily  than  the  motion  of  a  spot  of  light.  At  all  events,  it 
seemed  interesting  to  endeavor  to  register  the  excursions  of  the  horizontal 
pendulum  by  displacement  interferometry,  not  so  much  with  a  view  to  re- 
cording seismological  phenomena,  as  to  approach  by  this  means  certain 
other  problems,  the  tilting  of  the  earth's  surface  relatively  to  the  plumb-line, 
the  measurement  of  the  constant  of  gravitation,  etc.  The  present  paper,  there- 
fore, undertakes  a  new  departure  with  this  special  end  in  view,  with  possibly 
some  ulterior  bearing  on  microseismology. 

If  the  inclination  of  the  axis  of  the  horizontal  pendulum  is  but  a  few  degrees 
to  the  vertical  and  a  large  framework  is  in  question  (there  is  scarcely  any  limit 
to  size  other  than  strength  of  the  material),  the  sensitiveness  of  the  apparatus, 
when  the  excursions  are  read  off  in  terms  of  light-waves,  is  astonishing;  or  at 
least  it  would  be  so  if  the  instrument  supplied  with  mirror  and  screen  had  not 
been  so  thoroughly  perfected.  The  horizontal  pendulum,  moreover,  has  this 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      31 


peculiarity,  that  it  is  able  to  support  relatively  large  weights;  i.e.,  relatively 
massive  bodies  may  be  subjected  to  each  other's  attraction. 

19.  Apparatus. — The  horizontal  pendulum  has  the  usual  form  of  a  swing- 
ing gate  and  was  constructed  of  ^-inch  (vertical)  and  /^-inch  (oblique)  thin 
steel  tubes.  The  material  available  here  was  unfortunately  slightly  too  thick- 
walled,  a  defect  which  will  be  modified  in  the  future.  Moreover,  steel,  as  has 
been  seen  in  the  work  with  the  electrometer,  is  an  undesirable  metal  in  the 
varying  magnetic  field  of  a  city  when  the  micrometry  of  angles  is  in  question. 

The  frame  of  the  pendulum,  as  shown  in  fig.  19,  is  very  simple.  ABC  is  the 
truss  of  steel  tube,  soldered  at  A  and  B  and  terminating  in  the  brass  clutch  at 
C,  into  which  it  is  also  soldered. 
The  tube  AB  is  slotted  at  top  and 
bottom  and  each  end  receives  a 
solid  cylinder,  a  and  b,  of  glass- 
hard  steel,  snugly.  These  are  held 
in  place  by  collars  c  and  d.  The 
cylinder  b  contains  a  conical  socket 
to  receive  the  point  of  the  hori- 
zontal steel  pivot  t,  a  portion  of 
the  tube  A  having  been  removed 
at  this  part.  Similarly  the  cylin- 
der a  contains  a  vertical  slot  (or 
reentrant  dihedral  edge)  to  re- 
ceive the  horizontal  pivot  s. 
These  pivots  are  adjustable  to- 
ward and  from  the  rear,  from 
right  to  left,  and  each  is  revolv- 
able  about  a  horizontal  axis 
normal  to  the  figure,  in  a  way 
which  will  presently  be  shown. 
The  distance  between  pivots  was 
97  cm.,  the  distance  between  the 
cylinders  AB  and  D  about  1 1 1  cm.,  and  the  reduced  end  projects  about  16  cm. 
beyond  the  edge  EE  of  the  brick  pier  to  which  the  pivots  are  attached.  D, 
clutched  by  C,  is  the  hollow  stem  of  the  tablet  /,  which  holds  the  plane  dot 
slot  arrangement  to  secure  the  grating  g,  a  spring  passing  down  the  interior  of 
the  tube  D.  The  lower  pivot  /  should  preferably  point  towards  the  center  of 
gravity  G. 

The  whole  apparatus  is  inclosed  in  a  more  or  less  triangular  flat  case  h'mnk, 
firmly  bolted  to  the  wall  at  q,  m,  and  p.  The  two  sides  of  the  case  beyond  the 
pier,  h'ilk,  may  be  slid  off  to  the  left,  and  then  the  whole  remainder  lifted  off 
its  bearings  without  touching  the  pendulum,  as  the  case  has  no  rear  wall. 
The  front  face  is  within  3  inches  of  the  face  of  the  pier.  This  arrangement 
was  found  very  satisfactory.  The  head  of  the  case  ki  is  of  course  glass-faced 


FIG.  19. 


32     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

(identical  plates)  in  front  and  rear,  so  that  the  grating,  etc.,  may  be  seen.  A 
convex  mirror  if  placed  at  g  reflects  a  beam  of  light,  showing  the  pendulum  to 
be  nearly  stationary  during  the  day,  in  spite  of  the  surrounding  city.  The 
slow  normal  variations  were  not  greater  than  5  mm.  on  a  radius  of  13  meters, 
corresponding  therefore  to  about  40  seconds  of  arc.  The  corresponding  change 
of  inclination  relative  to  the  plumb-line  would  be  less  than  one  one-hundredth 
of  this,  depending  on  the  period  given  to  the  horizontal  pendulum. 

The  mass  of  the  pendulum  was  720  grams,  that  of  the  grating  holder  orig- 
inally 475  grams,  and  that  of  the  grating,  etc.,  about  55  grams,  making  a  total 
of  i ,2 50  grams ;  but  these  masses  are  to  be  much  modified  in  the  future.  The 
center  of  gravity,  at  G,  with  the  grating  in  place,  was  originally  about  80  cm. 
from  the  axis  AB. 

The  grating  at  g  moves  between  the  two  opaque  mirrors,  usually  called  M 
and  N,  of  the  displacement  interferometer,  in  the  way  shown  in  my  earlier 
work  on  interferometry. 

But  these  mirrors  M  and  N  must  in  the  present  case  be  identically  concave, 
silvered  on  their  front  faces,  and  at  a  distance  equal  to  their  common  radius 
of  curvature  from  the  center  of  the  ruled  face  of  the  grating.  This  center  is 
illuminated  by  the  impinging  beam  of  light  from  the  collimator,  and  the  re- 
turned beams,  reflected  from  M  and  N,  must  pass  through  the  same  area  of 
illumination.  In  such  a  case  the  reflection  at  M  and  AT  is  always  normal  to 
those  surfaces  and  the  rotation  of  the  grating  does  not  interfere  with  the  defi- 
nition of  the  ellipses  of  the  interference  pattern.  For  any  other  distance  of  M 
and  N,  except  these  radii  of  curvature,  the  spectra  in  the  telescope  will  cease 
to  coincide  horizontally  on  rotating  the  grating  and  the  ellipses  would  at  once 
vanish.  On  the  other  hand,  the  displacement  of  the  grating  in  arc  at  the  end 
of  the  arm  of  the  horizontal  pendulum  is  registered  in  amount  by  the  shifting  of 
the  ellipses  in  the  interfering  spectra.  This  displacement  includes,  of  course 
(as  a  small  correction),  the  additional  thickness  of  glass  introduced  by  the 
rotation  of  the  grating.  The  displacement  in  question  is  the  arc,  which,  when 
referred  to  the  axis  of  the  horizontal  pendulum,  measures  its  angular  devia- 
tion resulting  from  the  inclination  of  the  earth's  surface  relatively  to  the 
plumb-line. 

It  is  convenient  to  exhibit  the  details  of  the  instrument  (figs.  20  and  21)  in 
separate  parts  for  convenience  in  drawing,  these  being  superimposed  in 
practice. 

Fig.  20  shows  the  attachment  of  the  two  opaque  mirrors  M  and  N  of  the 
interferometer  to  the  pier  P.  Here  abed  is  an  ordinary  framework  of  X-inch 
gas-pipe.  The  end  a  is  firmly  plastered  into  the  pier,  b  rises  at  a  slight  angle, 
cd  being  horizontal  and  parallel  to  the  pencil  of  light  from  the  slit,  while  g 
shows  the  position  of  the  grating  on  the  horizontal  pendulum  in  fig.  19.  The 
arm  b  lies  below  the  case  in  that  figure  and  is  free  from  it.  Each  of  the  mir- 
rors M  and  N  is  on  plane  dot  slot  adjustments,  and  M  is  provided  with  a 
Fraunhofer  micrometer  suggested  in  the  figure.  Both  M  and  AT  can  be  rotated 
around  horizontal  and  vertical  axes  for  adjustment,  the  former  M  being  pro- 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      33 

vided  with  a  fine  motion.  The  clutch  e  and  the  corresponding  one  for  M 
(not  shown)  allow  the  micrometer  to  be  placed  at  a  greater  or  less  distance 
from  the  grating.  The  center  of  the  mirror  is  about  on  the  same  horizontal 
level  as  the  grating.  It  is  also  usually  convenient  to  place  the  lens  L  of  the 
collimator  in  its  screen,  on  the  same  rod  cd,  with  an  appropriate  clutch  and 
rack  and  pinion. 

The  complementary  framework  is  shown  in  fig.  21  and  holds  the  slit  5  of 
the  collimator  (or  the  filament  of  a  Nernst  lamp)  and  the  two  telescopes  Tand 
T'  in  place  for  observation,  T  being  used  for  the  direct  slit  image  and  T'  for 
the  diffraction  spectra  and  interferences.  The  framework  fghi  is,  as  before,[jof 


FIG.  ao. 


FIG.  21. 


gas-pipe,  /  being  firmly  plastered  into  the  wall  on  the  front  face  of  the  pier 
(the  other  one,  ab,  being  on  the  side).  The  telescopes  T  and  T'  are  necessarily 
adjustable  on  a  horizontal  and  vertical  axis,  and  may  be  raised  and  lowered 
and  moved  right  and  left  along  the  rod  kl,  held  by  a  firm  clutch  at  k.  The 
lens  L  may  also  be  carried  on  ab,  as  has  been  stated.  Right-and-left,  up-and- 
down  motion  is  needed  for  the  insertion  of  these  appurtenances.  The  rods  cd 
and  hi  are  not  in  the  same  horizontal  or  the  same  vertical  plane,  so  that  the 
systems  may  be  superposed  as  stated. 

In  the  course  of  the  work  it  appeared,  however,  that  the  framework  of 
simple  pipe  was  annoyingly  subject  to  tremors.  It  was  found  necessary  to 
lengthen  the  rods  gd  and  gc,  g'i  and  g'h  to  over  a  meter  in  length.  Hence  it 
was  preferable  to  bolt  the  pairs  of  parallel  rails  together  for  increased  stiffness 
and  to  secure  the  ends  of  each  pair  with  a  wide  tetrahedral  brace  of  gas-pipe, 
abutting  at  the  pier,  against  horizontal  and  vertical  displacement.  So  ad- 
justed, the  system  was  light  and  rigid  and  easily  modified  for  the  different 


34     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

purposes  of  the  experiment.  The  additional  braces  have  not  been  shown  in 
the  figure,  as  they  depend  on  purely  local  conditions,  the  base  of  each  tetra- 
hedron being  at  the  pier  and  its  apex  at  the  corresponding  common  ends  of 
the  pairs  of  rails,  gc,  g'h,  g'i,  and  gd.  All  appurtenances  like  lenses,  mirrors 
and  micrometers  are  attached  with  strong  removable  clamps,  provided  where 
needed  with  rack-and-pinion  attachment  for  focussing,  etc.  This  allows  of 
an  easy  and  indefinite  modification  of  the  sytem  and  is  thus  very  convenient 
for  experimental  purposes  of  the  present  kind.  In  the  later  work  the  telescope 
rod  kl,  fig.  21,  was  discarded  in  favor  of  a  tripod  standing  on  the  floor. 

It  is  finally  necessary  to  describe  the  pivots  of  the  horizontal  pendulum, 
and  these  are  also  given  in  fig.  20.  Here  p  is  a  length  of  ^(-inch  gas-pipe  fixed 
in  the  wall  with  plaster.  The  outer  end  is  split  lengthwise  and  carries  a  collar 
and  set-screw  I,  so  that  the  brass  rod  q  fitting  the  pipe  p  snugly  may  be  firmly 
secured.  The  end  of  q  carries  the  horizontal,  very  snugly  fitting  screw  m  of 
^-inch  brass,  which  is  tipped  at  n  with  the  steel  point  of  a  darning  needle. 
The  point  of  n  is  received  by  the  socket  of  the  horizontal  pendulum.  Thus 
n  may  be  rotated  about  qp  and  moved  fore  and  aft  or  right  and  left  for  ad- 
justment. The  socket  is  a  conical  hollow  of  about  60°  and  of  glass-hard  steel. 

20.  Equations.  —  With  regard  to  the  apparatus  just  described,  the  size  of 
which  was  limited  to  conveniently  fit  the  given  pier,  the  following  equations 
may  be  used  to  obtain  an  estimate  of  the  sensitiveness  to  be  expected. 

Let  <f>  be  the  inclination  of  the  axis  of  the  pendulum  to  the  vertical  and  6  an 
angular  excursion  of  the  pendulum,  measured  from  its  position  of  equilibrium. 
Let  h  be  the  normal  distance  of  the  center  of  gravity  from  the  axis.  The  rise 
of  the  latter  above  its  lowest  position  is 

w  ;y=/t(i—  cos  5)  sin  ^= 


and  the  energy  potentialized,  if  the  total  mass  is  M,  will  be 


which  for  small  displacements  corresponds  to  the  torque  Fh  at  the  angle  6. 
This  torque  is 

(3)  -~-  =  Mgh  sin  <p  sin  6  =  Mgh  <pe  nearly 

or  the  total  force  F  acting  at  center  of  gravity,  or  F/M  per  gram  of  mass  M  is 


In  the  above  apparatus  M=  1,245  grams,  /*  =  8o  cm.    Hence  per  vanishing 
interference  ring,  since  the  grating  moves,  if  AA/"  is  the  displacement  of  the 
micrometer  to  bring  back  the  center  of  ellipses  to  the  fiducial  sodium  line 
(5)  „ 


where  R  is  the  distance  of  the  point  of  the  grating  at  the  line  of  light  corre- 
sponding to  the  slit,  from  the  axis  of  rotation.    In  the  apparatus  R  =  1  1  1  cm. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     35 
Hence  the  angle  corresponding  to  a  vanishing  ring  is,  since  AAf=3oXio~*, 
6  =  ^r^  =  13  X  io-8  radian  =  0.028* 

Furthermore,  if  <p=i°  =  0.0175  radian 

F/M=98iX.oi7sXi3Xio-8=2.3Xio-8  dyne 

per  vanishing  ring  per  gram-mass  at  the  center  of  gravity  of  the  pendulum. 
The  total  pull  of  the  center  of  gravity  of  the  above  pendulum  is  thus 

F=  2.3  Xio-^X  1245  =  2.9X10-*  dyne 

per  vanishing  ring,  on  one  side.  By  lengthening  the  radius  from  h  to  R  this 
may  be  decreased  to  about  2Xio~8  or  less.  Hence  in  case  of  gravitational 
attraction  at  one  centimeter  of  distance  it  would  require  two  equal  masses  m 
(since  7  =  6.7  X  io-8  roughly)  of  the  value  m  =  \/2o  X  io4/6.7  =  i  .8  X  io2grams, 
or  180  grams  per  vanishing  interference  ring,  at  a  distance  of  i  cm.  On  the 
other  hand,  the  framework  of  the  above  pendulum  is  unnecessarily  heavy,  and 
was  constructed  out  of  the  material  at  hand.  It  could  easily  be  reduced  in 
weight  much  below  the  above  datum,  or  the  greater  part  supported  on  a  float, 
so  that  the  case  may  be  stated  many  times  more  favorably. 

Resuming  equation  (3),  if  AT  is  the  moment  of  inertia,  i  the  radius  of  gyra- 
tion, and  T  the  period  and  /  the  length  of  the  horizontal  pendulum, 

(6)  T=° 

an  equation  from  which  <p  may  be  found  in  terms  of  T,  i,  and  h  which  must  be 
measured. 

Again,  the  indicated  length  H  of  pendulum  (distance  from  the  center  of 
gravity  to  the  point  of  intersection  of  the  axis  and  the  plumb-line  through 
the  center  of  gravity)  is 

(7)  H  =  Jt/sin  <f>  =  h/<p,  nearly . 

The  change  of  vertical  inclination  a  of  the  axis  of  the  pendulum  correspond- 
ing to  the  horizontal  deviation  6  is,  then, 

hd 

(8)  a  =-jj=e<f>>  nearly, 

or  if  the  period  T  be  introduced  from  (6)  and  6  from  (5) 

4**?    AAT 

(9)  a=ng   1R 

It  is  in  equation  (8)  that  the  condition  of  remarkable  sensitiveness  resides. 
Thus,  if  the  interferometer  is  used,  a  =  <f>AN/2R,  and,  if  A/V= 30X10-'  and 
v?  =  io~2  (somewhat  less  than  i°  of  arc),  R=  in  cm.,  as  above, 

a  =  13  X  io-10  radians  =  •  00028" 
per  vanishing  interference  ring. 


36     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

If  F,  as  before,  is  the  force  at  the  center  of  gravity,  the  corresponding  force 
at  the  grating,  a  distance  R  from  the  center,  is 

(10)  FR -  Mg<fA-£  =  M-jjgd 

since  <f>  is  given  by  equation  (6).  This  equation  implicitly  contains  h,  since  * 
refers  to  an  eccentric  axis  and  iz=i\+h'i;  but  i  may  be  found  directly. 

The  deviation  0  is  given  by  (5).  If,  however,  the  device*  of  two  parallel 
mirrors,  equidistant  (distance  R)  from  the  axis  of  the  horizontal  pendulum, 
be  used,  and  if  light  impinges  on  either  mirror  at  an  angle  of  incidence  I  (the 
impinging  and  reflected  beams  being  always  parallel), 

AAT 

where  AN7  is  the  displacement  of  the  micrometer.  The  horizontal  pendulum 
is  in  this  case  constructed  symmetrically  to  the  vertical  axis  in  the  form  of  a 
balance  beam,  but  somewhat  heavier  on  one  side. 

Finally,  the  compound  pendulum  may  be  supported  on  a  cylindrical  float, 
symmetrical  or  not  to  the  vertical  axis  of  the  pendulum  and  submerged  in 
water  or  some  other  liquid.  In  such  a  case,  the  mass  of  the  compound  pen- 
dulum may  be  reduced  in  any  degree  without  serious  difficulty  from  capillary 
forces,  as  will  be  shown  below.  If  the  center  of  buoyancy  is  in  the  vertical 
line  passing  through  the  center  of  gravity  of  the  horizontal  pendulum,  the 
above  equation  needs  but  slight  alteration.  Let  V  be  the  volume  of  the  float, 
so  that  Vpg  is  the  buoyancy.  Apart  from  the  temperature  conditions,  p=  i, 
and  hence  the  equations  take  the  successive  forms,  since  (M—  V)g  is  sup- 
ported at  the  center  of  gravity,  instead  of  Mg, 


(13)  r-(Jlf 

The  force  at  a  distance  R  from  the  axis  is,  when  the  center  of  gravity  is  at  a 

distance  h, 


Hence  the  force  has  been  reduced  in  the  ratio  of  M/(M-  V}  for  the  same  6. 
One  may  also  note  that  it  is  smaller,  not  only  as  <p  is  smaller,  but  as  h/R  is 
smaller.  Hence  a  symmetrical  form  of  pendulum,  like  the  balance-beam,  but 
slightly  heavier  on  one  side,  suggests  itself  for  work  on  gravitational  attrac- 
tion, etc.  It  was  not  found  difficult  to  reduce  the  weight  of  the  pendulum  by 
flotation  to  40  grams,  i.e.,  about  31  times.  Hence  the  force  per  vanishing 
interference  ring  computed  above  would  now  be 

F'R  =>  2  X  io-5/3  1  =  6  X  lo-6  dynes,  roughly. 

This  would  be  equivalent  to  the  attraction  of  two  so-gram  weights  at  i  cm. 
of  distance. 

*Barus:  Am.  Journ.  Sci.,  xxxvii,  pp.  83  et  seq.,  1914. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      37 
Furthermore, 


whence,  since  0=&N/2R 

f  a  x"      M  ****»    M 

('6)  ^-K-T^-tf 

all  of  which  quantities  are  easily  determined  with  accuracy.  To  find  the 
radius  of  gyration  *',  for  instance,  a  body  of  known  moment  of  inertia  may  be 
suspended  at  the  end  of  the  horizontal  pendulum  and  the  periods  T  of  the 
pendulum  before  and  after  suspension  determined,  with  or  without  the  float. 
Finally  the  change  of  vertical  inclination  a  becomes 


If  the  pendulum  is  damped,  which  will  usually  be  the  case,  it  may  be  neces- 
sary to  observe  the  logarithmic  decrement,  in  order  to  compute  the  free  period 
in  the  usual  way. 

If  the  buoyant  force  due  to  the  float  does  not  pass  through  the  center  of 
gravity  of  the  solid  parts  of  the  pendulum,  but  at  a  distance  h'  from  the  ver- 
tical or  pivotal  axis,  the  new  distance  of  the  center  of  gravity  h"  when  the 
pendulum  is  partially  floating  will  be 


if_ 


M-V 

Hence,  tih'=h,  then  h"=h,  resulting  in  the  equations  just  deduced.    But  if 
h'  =  o,  i.e.,  if  the  buoyant  force  passes  through  the  point  of  the  lower  pivot, 


Thus  the  equations  deduced  become  identical  with  the  original  equations  (2) 
ft  seq.  The  float  therefore  adds  nothing  to  the  sensitiveness  except  in  so  far 
as  it  removes  friction  at  the  pivots  and  supplies  a  reliable  damper  for  the  pen- 
dulum. It  is  in  this  form  that  the  float  will  be  applied  below.  Since  the 
torque  equation  is  now  again 


where  all  references  are  to  solid  parts  of  the  pendulum,  h  may  be  accurately 
found  by  placing  weight  m  at  a  distance  /  from  the  plane  of  the  pendulum,  or 
better,  by  placing  weights  alternately  before  and  behind  this  plane,  at  a  dis- 
tance /  apart.  The  torque  applied  is  then  T=mgl,  whence 

ml 
<I9)  h=Mi 

This  method  will  be  used  effectively  in  several  experiments  below.  It  is  an 
excellent  test  on  the  reliability  of  the  damper,  since  h  can  also  be  determined 
directly  by  the  suspension  of  the  solid  beam  of  the  pendulum.  In  the  adjust- 
ment adopted,  at  a  scale  distance  of  900  cm.,  m/=gramXcm.  on  the  scale- 
pan,  produced  a  deflection  of  about  i  mm. 


20921O 


38     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

A  few  other  equations  of  minor  importance  may  be  added.  If  the  indicated 
length  is  H  and  the  horizontal  pendulum  be  treated  as  a  vertical  pendulum 
of  length  L,  the  point  of  suspension  being  the  intersection  of  the  plumb-line 
through  the  center  of  gravity  and  the  line  determined  by  the  points  of  the 
two  pivots,  the  observed  period  is 
(20)  r=WZ/g  and  P=LH 

where  7  is  the  corresponding  radius  of  gyration. 

If  the  end  of  the  horizontal  pendulum  is  loaded  with  the  weight  m  of  a 
disk  at  a  mean  distance  R  from  the  axis  for  the  measurement  of  gravitational 
attraction,  since  (M+m)  h'=Mh+mR,  the  new  force  at  R  is 


When  the  end  of  the  pendulum  is  similarly  loaded  for  the  determination  of 

its  radius  of  gyration,  since 

(aa) 

the  new  period  is 


r=     v 

V  Tj_™  _ 

h 

Since  T'  and  T  are  observed  and  m,  M,  R,  h  given,  *  may  be  computed. 
The  horizontal  pendulum  itself  thus  supplies  the  value  of  i. 

If  the  lower  pivot  is  provided  with  a  strong  micrometer  screw,  by  which  it 
may  be  moved  over  a  small  distance  z  to  the  front  or  rear  of  the  plane  of  the 
pendulum,  the  computed  value  of  a  may  be  tested  independently.  Thus  let 
y  be  the  distance  apart  of  the  pivots  and  z  the  displacement  of  the  lower,  then 

when  in  the  method  of  deflection  x  is  the  increase  of  the  distance  apart  of  the 
two  images  of  the  slit,  at  a  distance  D  from  the  further  mirror.  Hence 

where  <p  must  agree  with  its  corresponding  datum  from  the  pendulum  meas- 
urement in  terms  of  period.  Thus,  since  <f>  and  h  may  be  obtained  inde- 
pendently, the  torque  T,  etc.,  is  given  independently.  This  method  will  also 
be  applied  below. 

21.  Observations  with  a  grating  rotating  on  a  fixed  vertical  axis.— When 
the  opaque  mirrors  M  and  N  are  identically  concave  and  are  put  on  the  ordi- 
nary interferometer  at  a  distance  equal  to  their  radius  of  curvature  from  the 
stationary  grating,  the  latter  may  be  rotated  (without  translation)  as  far  as 
the  breadth  of  the  opaque  mirror  N  permits,  without  readjustment.  The 
ellipses  are  not  lost.  Inasmuch,  however,  as  different  thicknesses  of  glass  are 
introduced  into  the  rays  when  the  grating  is  rotated,  the  ellipses  travel  hori- 
zontally through  the  spectrum  from  the  red  to  the  violet  end  or  the  reverse. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     39 

They  are  about  equally  clear  in  all  positions.  A  displacement  at  the  mirror 
N  of  about  4  cm.  per  meter,  i.e.,  0.04  radian,  equivalent  to  a  rotation  of  2.3°  of 
the  reflected  ray,  or  a  rotation  of  1.15°  for  the  grating,  was  within  the  scope 
of  the  interferometer  and  the  tests  were  made  within  this  limit.  It  is  far  in 
excess  of  anything  required  in  the  horizontal  pendulum.  No  doubt  if  the 
mirror  N  had  been  wider,  the  ellipses  could  have  been  retained  for  larger  angles 
of  rotation  of  the  grating,  though  they  would  in  such  a  case  travel  several 
times  through  the  spectrum.  The  micrometer  at  M  would  have  to  be  used. 
If  long  columns  of  glass  are  to  be  inserted  in  either  beam  (GM  or  GN)  the 
concave  mirror  is  not  available,  since  the  direct  slit  images  will  then  have 
different  focal  positions.  The  rays  issue  from  the  plane-parallel  column, 
parallel  to  this  focal  direction,  but  from  a  virtual  focus  nearer  the  concave 
mirrors.  Hence,  if  the  column  is  placed  in  the  beam  GM,  the  beam  GN  will, 
as  a  rule,  have  to  be  correspondingly  shortened.  The  algebraic  relations  are 
complicated. 

22.  Observations  with  the  interferometer.  —  The  horizontal  pendulum  with 
which  the  following  observations  were  made  had  the  following  constants,  M 
being  the  total  mass  of  the  fixed  parts,  m  the  attached  mass,  h  the  distance 
of  the  center  of  gravity  from  the  axis,  R  the  distance  of  the  vertical  line  of 
light  on  the  grating  (also  mean  distance  of  m  and  of  FR)  from  the  axis,  <p  the 
inclination  of  the  axis  :  ^=1,250  grams;  #1  =  227  grams;  h  =  8o  cm.;.R  =  iii.3 
cm.  The  observed  periods  (primes  refer  to  the  loaded  pendulum)  for  M  and 
M+m  were  7=18.48  seconds;  T'  =  18.87  seconds.  Thus  ^  =  85.1  cm.; 

tf>  =  a/6=o.QioBi  radian=o.62° 

and  H  =  7,394  cm.;  L=8,488cm.;  #'=7,834  cm.;  £,'  =  8,853  cm. 

Since  6=AN/2R  when  AN  is  the  mean  displacement  for  the  horizontal 
deflection  (6)  of  the  pendulum, 

a  =  i  o-5  X  4.  86  AAf  radians. 

Thus,  if  AAf  =  10—  4  cm.,  a  =  io~3  second  of  arc,  or  the  change  of  a  per  vanish- 
ing interference  ring  (AAf  =  10-^X30)  is  0.000310  second  of  arc.  Since  T 
may  easily  be  increased  over  3  times,  this  limit  may  be  reduced  to  a  =  .000030" 
per  ring. 

Similarly,  the  forces  at  distance  R  from  the  axis  of  the  horizontal  pendulum 
are 


Thus  if  AAT=  io~4  cm.,  F'#  =  o.oo54  dyne  or  about  0.0016  dyne  per  vanishing 
interference  ring,  in  case  of  the  pendulom  loaded  with  the  disk  m. 

In  the  graph  which  follows  an  example  is  given  of  a  series  of  observations 
made  for  6  and  a,  and  no  further  explanation  will  be  needed.  Since 
a  =<pO  =  o.oio80,  a  need  not  be  recorded. 


40     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

In  fig  22  these  observations  have  been  inscribed,  the  ordinates  being  the 
inclination  of  the  pier  a,  in  hundredths  of  a  second  of  arc,  very  nearly.  It 
will  be  seen  that  the  inclination  increases  as  a  whole  from  the  beginning  to 
the  end  of  the  month,  the  total  range  lying  within  something  over  2  seconds 
of  arc.  The  rise  is  particularly  marked  and  sustained  after  the  i4th,  and  the 
difference  of  inclination  between  the  first  and  second  halves  of  the  month  is 
about  i  second. 


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W&   6        67        t       9       *>       11       &       ft       M       15       16      1T 

FIG.  22. 

As  the  observations  were  made  in  an  unavoidably  steam-heated  room,  it 
is  probable  that  the  flexure  of  the  pier,  etc.,  due  to  thermal  causes,  has  been 
largely  operative  in  modifying  the  trend  of  the  curve;  for  on  comparing  the 
curve  as  a  whole  with  the  thermostat  sheets  (not  shown)  a  retarded  effect  is 
possibly  suggested,  such  as  one  would  suspect  if  variations  of  surface  tem- 
perature should  penetrate  massive  masonry.  It  would  then  be  possible  for 
the  curve  to  have  different  heights  at  the  same  temperature.  Naturally  such 
comparisons  are  very  vague,  and  it  is  the  range  of  values  of  a  admissible  in 
the  apparatus  which  is  here  of  paramount  interest.  Furthermore,  as  the  hill 
on  which  the  laboratory  stands  is,  at  present,  being  tunneled,  so  that  the 
building  is  subject  once  or  twice  a  day  to  the  tremors  resulting  from  the 
vigorous  blasting  underground,  adequate  conditions  for  the  installation  of  an 
apparatus  of  the  present  kind  are  still  remote.  It  is  really  surprising  that 
interferometer  observations  could  be  made,  without  essential  difficulty,  under 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      41 

these  circumstances.  During  an  explosion,  of  course,  the  ellipses  vanish,  to 
reappear,  however,  immediately  afterward,  sometimes  with  displacement,  such, 
for  instance,  as  is  indicated  by  the  doubled  parts  of  the  curve.  The  use  of  the 
water  damper,  moreover,  which  was  necessary  here,  is  objectionable,  though 
it  has  not,  probably,  introduced  any  marked  error  into  the  observed  curve 
(see  doubled  parts).  Finally,  the  use  of  a  steel  horizontal  pendulum  with  its 
plane  in  the  magnetic  meridian  is  inadmissible.  I  have  not,  therefore,  endeav- 
ored to  interpret  the  results,  but  they  are  given  simply  as  an  example  of  a 
systematic  series  of  observations,  extending  over  a  month.  I  hope  in  the 
summer  to  resume  the  work  in  the  absence  of  the  annoyances  referred  to. 

I  may  add  in  conclusion  that  the  experiments  referred  to  above,  for  measur- 
ing the  gravitational  attraction  of  two  identical  brass  disks,  led  to  curious 
results.  It  is  easily  seen  that  for  constant  mass  the  attraction  of  nearly  con- 
tiguous disks  should  increase  roughly  as  the  fourth  power  of  their  radius. 
For  disks  20  cm.  in  diameter,  however,  the  result  is  an  invariable  repulsion, 
several  times  as  large  as  the  estimated  gravitational  attraction,  the  position 
of  equilibrium  being  reached  gradually  in  the  lapse  of  several  minutes.  The 
subject  will  be  systematically  discussed  in  Chapter  II. 

23.  Further  observations.  Film  grating.  Oil  damper. — After  the  above 
experiment,  the  steel  horizontal  pendulum  was  used  for  other  purposes  and 
observations  on  the  tilting  of  the  pier  were  discontinued.  Later,  however, 
the  apparatus  was  again  available  and  a  variety  of  experiments  was  made 
with  it.  In  the  first  place,  the  water  damper  was  replaced  by  an  oil  damper, 
as  it  seemed  probable  that  the  surface  tension  of  illuminating  oil  and  its  slower 
evaporation  would  be  an  advantage.  Under  like  conditions  (though  it  proved 
sufficiently  serviceable)  it  did  not  check  the  vibration  as  effectively  as  the 
water  damper.  The  modification  of  chief  interest,  however,  was  the  inser- 
tion of  one  of  Mr.  Ives's  film  gratings  (in  the  usual  double  plate-glass  pro- 
tection) in  place  of  the  plate-glass  grating.  The  film  grating  in  question  had 
about  15,000  lines  to  the  inch,  so  that  the  dispersion  was  excessive,  the  ellipses 
being  large  and  diffuse  and  with  a  long  horizontal  axis.  To  obviate  this  diffi- 
culty a  thick  compensator  was  introduced  into  the  component  beam  M  passing 
to  and  from  the  micrometer.  For  this  purpose  three  thick  plates  of  glass 
were  cemented  together  with  Canada  balsam  to  a  combined  thickness  of 
something  over  2  cm.  The  ellipses  now  became  adequately  sharp  and  almost 
circular  in  form.  In  consequence  of  the  multiple  reflections  described  in  Chap- 
ter IV,  Part  II,  the  ellipses  are  not  so  strong  as  in  case  of  the  grating  ruled 
on  plate  glass,  and  they  are  much  harder  to  find;  but  they  are  nevertheless 
quite  serviceable.  The  single-plate  film  grating  of  §  60  was  not  at  hand  at  the 
time.  It  is  advisable  to  try  out  the  double-plate  film  grating  first  on  the  fixed 
interferometer,  in  order  to  determine  which  lines  of  the  individual  images  of 
the  slits  are  to  be  placed  in  horizontal  and  vertical  contact,  together  with  the 
distance  which  corresponds  to  the  different  interferences  on  the  micrometer. 
After  this  is  done,  the  corresponding  adjustment  of  the  interferometer  is  easier. 


42     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

It  is  also  advisable  to  adjust  the  plate-glass  grating  in  both  cases  for  compar- 
ison. In  figs.  23,  24,  25,  26  an  example  is  given  of  these  observations  in  the 
usual  way.  Data  between  January  15  and  24  (fig.  23,  A)  and  March  3  and  6 
(preceding  fig.  23,  B)  exhibit  the  behavior  of  an  oil  damper  with  the  ruled 
grating.  In  the  observations  after  March  14,  running  as  far  as  July  14,  1914, 
the  ruled  grating  was  replaced  by  the  film  grating.  Inasmuch  as  no  essential 
change  was  made  at  the  steel  horizontal  pendulum,  the  constants  may  be 
taken  to  be  the  same  as  above,  viz,  M=i,2so  grams;  h  =  8ocm.;  R  =  m  cm.; 
7=18.48  sec.;  *  =  8s  cm.;  ^=0.0108  rad.  =  0.62°.  Thus  a  =  io-6X48.6  AN 
rad.  =  loAN",  nearly. 


24.  Inferences. — The  curves  in  fig.  23,  A,  B,  are  independent  so  far  as  zero 
of  measurement  is  concerned,  but  they  already  exhibit  a  tendency  to  decline 
in  the  direction  of  a  decrease  of  a.  This  was  pronounced  in  January  and  also 
in  March.  It  is  not  a  regular  decrease,  so  that  the  cause  can  hardly  be,  or  at 
least  not  wholly  be,  sought  in  the  yield  of  the  parts  of  the  apparatus;  for  in 
such  a  case  there  would  be  no  recovery  (increase  of  a),  a  feature  which  is  often 
marked.  The  continuous  observations  (i.e.,  with  the  same  uninterrupted  zero) 
are  given  in  figs.  24,  25,  26.  The  same  scale  is  used  throughout,  but  on  April 
22  and  May  15  it  was  necessary  to  displace  the  graph  in  order  to  accommo- 
date the  observations  on  the  sheet.  The  amount  of  displacement  is  shown. 
Here  also  there  is  a  gradual  and  continuous  decrease  of  the  values  of  a.  Begin- 
ning on  April  i  with  a  about  2",  the  observations  pass  through  a  succession 
of  oscillations  to  the  lowest  value  of  a  recorded,  about  -4.2",  on  May  18. 
After  this  there  is  intermittent  partial  recovery,  so  that  on  June  28  a  has 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      43 


risen  to  -i".  The  decline,  however,  at  once  commences,  and  on  July  8  a  is 
about  -3.3",  from  which  it  rises  to  -2.4"  at  the  end  of  the  work.  The  maxi- 
mum range  of  the  tilting  of  the  pier,  a,  between  April  and  July  is  thus  about 
from  +2"  to  -4",  or  6"  of  arc. 


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44     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

Between  the  oscillations  proper,  which  are  as  a  rule  sharply  marked,  both 
at  the  maxima  and  the  minima,  there  are  regions  of  relative  constancy  of 
inclination.  Thus  between  April  9  and  21  the  inclination  is  nearly  constant 
and  about  0.6";  between  May  17  and  28  there  is  a  slow  ascent  from  about 
—  4.1"  to  —3.6";  between  June  i  and  12  a  slow  descent  from  —3.0"  to 
-3!4";  between  June  16  and  25,  relative  constancy  at  about  -2.8",  etc. 
Regarding  the  observations  as  a  whole,  it  is  not  impossible  that  there  may 
have  been  some  slow  yielding  (quiescent  frictional  forces  probably  yield  vis- 
cously) of  the  mechanical  and  the  optical  parts  of  the  apparatus.  The  main 
features  of  the  diagram,  however,  are  due  to  the  pier  itself,  or  the  pendulum, 
in  responding  to  actual  forces.  On  these  the  former  errors  may  have  been 


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PIG. 

SOtfuhZ       4        6        8        A?       «  '    •«•       /d 
26. 

superimposed.  Certain  large  drops  on  May  15  remain  unexplained.  The 
decline  after  April  3  is  large  and  slow,  so  that  it  could  be  observed  during  its 
occurrence.  This  may  therefore  have  been  actual  and  not  due  to  displace- 
ments of  the  pendulum  resulting  from  subterranean  shocks. 

The  observations  were  continued  into  June  and  July,  with  the  expectation 
that  when  the  basement  room  was  no  longer  heated,  the  variation  of  a  would 
practically  vanish;  but  this  is  not  at  all  the  case,  as  the  play  of  a  i.i  June  and 
July  scarcely  differs  from  the  average  run  of  values  during  the  winter  months. 

From  the  long-range  point  of  view,  inclination,  a,  decreases  from  about 
April  i  to  about  May  19,  after  which  it  increases  intermittently  again,  recov- 
ering on  June  29  and  again  on  July  27  (not  shown),  about  one-half  the  total 
decrement.  It  would  not  be  difficult  to  arrange  the  minima  in  semi-monthly 
periods,  if  any  reason  for  such  large  variations  of  a  could  be  assigned.  They 
are  of  course  enormously  above  anything  to  be  anticipated  from  tidal  influ- 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      45 

ences.  So  the  maxima  could  be  placed  at  March  17,  April  — ,  May  9,  June 
IS.  July— » March  29,  April  27,  May  29,  June  28,  July  27,  in  which  the  monthly 
periods  are  at  least  marked.  But  here  again  any  adequate  cause  for  such 
behavior  has  not  been  found,  nor  in  any  case  has  it  been  possible  to  separate 
the  true  from  the  adventitious  tilting  record. 

As  the  pendulum  was  a  thin  steel  tube  and  the  direction  north-south, 
one  might  infer  changes  of  the  earth's  horizontal  intensity.  It  is  hardly 
probable,  however,  since  whatever  magnetization  was  present  was  induced  by 
the  earth,  that  forces  of  the  required  intensity  could  be  present.  The 
mechanical  force  at  the  grating  for  the  displacement  AAf  would  be,  roughly, 
F=43AAf.  Since  i"  of  arc  of  a  was  about  ioAAf,  the  mechanical  force  in 
question  is  thus  4.3  dynes  per  second  of  a.  Such  forces  are  not  liable  to  be 
of  magnetic  origin. 

Finally,  if  we  compare  the  run  of  air  temperatures  given  after  May  28,  for 
instance  (the  thermostat  sheets  were  not  accurate  enough),  though  there  is  no 
detailed  resemblance  in  the  two  graphs,  some  relation  is  none  the  less  apparent. 
Thus  the  fall  of  temperature  up  to  June  10  and  its  rise  through  a  maximum 
on  June  14,  to  fall  again  to  June  22,  is  followed  by  the  pendulum  graph  with  a 
lag.  So  also  the  next  temperature  maximum  on  June  27  is  followed  by  a 
pendulum  maximum.  This  lagging  of  the  inclination  of  the  massive  pier  is 
precisely  what  one  should  expect  if  the  observed  oscillations  are  of  thermal 
origin.  It  would  seem  that  the  parts  of  the  pier  exposed  to  the  light  expand 
and  contract  on  the  more  equally  temperatured  colder  parts,  as  an  axis,  as  it 
were.  The  result  would  be  a  pendulum  mechanism,  very  similar  to  the  trian- 
gular bracket  which  I  have  discussed  above,  §  13 ,  and  which  is  peculiarly  sensi- 
tive to  the  elongation  of  its  parts.  The  expansion  of  any  side  of  a  triangle 
produces  relatively  marked  tilting  of  the  axis  when  the  instrument  of  detection 
is  a  horizontal  pendulum. 

Taking  the  observations  as  a  whole,  there  seems  thus  to  be  very  little 
opportunity  in  the  case  of  an  ordinary  massive  pier  of  conducting  observations, 
when  fixity  of  inclination  within  i"  of  arc  is  in  question,  even  for  brief  periods 
of  time.  Thus  even  after  June  28,  in  case  of  the  observed  pier,  there  are 
changes  of  a  amounting  to  2"  of  arc  in  ten  days,  and  0.2"  of  arc  per  day  must 
be  looked  upon  as  no  unusual  occurrence. 

25.  Improved  aluminum  pendulum.  Observations.— The  outstanding  ques- 
tion bearing  on  the  above  observations  was  the  possibility  of  a  magnetic  influ- 
ence in  case  of  the  horizontal  pendolum  made  of  steel  tubing,  the  pendulum 
being  otherwise  admirable  because  of  its  relative  strength.  A  new  pendulum, 
built  entirely  of  aluminum  tubing,  with  the  exception  of  the  brass  clutch  and 
the  vertical  hard-steel  bearings  for  the  pivots,  was  therefore  installed.  The 
aluminum  tubes  were  screwed  firmly  together,  the  large  triangle  having  the 
following  dimensions  and  constants:  Mass  of  pendulum,  554  grams;  mass  of 
grating  holder  and  leveler,  456  grams;  mass  of  (single-plate)  film  grating,  114 
grams;  mass  of  damper,  60  grams.  This  brings  the  total  weight  up  to  1,124 


46     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

grams,  not  differing  much  from  the  above.  If  lightness  is  an  object  (small 
torques  being  in  question),  the  clutch  and  grating  holder  should  also  be  made 
of  aluminum  and  a  lighter  grating  attached  ;  but  this  is  a  secondary  considera- 
tion here,  though  the  mass  might  easily  be  brought  down  to  about  700  grams. 

The  distance  of  the  center  of  gravity  from  the  axis  was  11  =  93.1  cm.;  dis- 
tance of  line  of  light  at  grating  from  the  axis,  R=no  cm.  The  period  was 
about  as  above,  T—ig  sec.  The  distance  apart  of  the  pivots,  £  =  97.1  cm. 

Though,  in  general,  the  aluminum  triangle  was  a  copy  of  the  steel  triangle, 
some  improvements  in  construction  were  introduced.  Thus  a  micrometer 
attachment  was  added  to  the  lower  pivot,  so  that  a  direct  value  of  <f>,  the  incli- 
nation of  the  pendulum  axis,  could  be  obtained.  Windows  were  put  in  the 
case  and  both  pivots  were  now  accessible  without  removing  it.  The  micrometer 
did  not  work  as  well  as  was  expected,  for  reasons  which  did  not  appear.  In 
several  series  of  experiments,  the  mean  of  the  horizontal  angle  corresponding 
to  30°  of  rotation  of  the  micrometer  screw  of  32  threads  to  the  inch  and  a 
distance  of  97.1  cm.  between  pivots  was 

6=  j~  =0.00793  radian 

since  the  reflected  spot  of  light  traveled  6.5  cm.,  when  the  scale  distance  was 
410  cm.,  for  each  step  of  30°  of  the  micrometer  screw.  The  corresponding 
change  of  inclination  of  the  pendulum  axis  would  correspond  to  one-twelfth 
of  the  pitch  of  the  screw  and  would  be 

-I?  ^=68.,X.o-  radian 

Since  a  =  <p6, 


not  differing  much  from  the  corresponding  value  in  case  of  the  steel  pendulum 
and  there  found  by  oscillation  measurements,  the  pivots  having  been  replaced 
as  nearly  as  possible  in  their  former  positions.  The  projected  horizontal  dis- 
tance apart  of  the  pivots  is  thus  about 

97.iX8.6Xio-3  =  o.83  cm. 

which  could  easily  be  decreased  and  the  pendulum  made  more  sensitive  (pos- 
ibly  ten  times).  Moreover,  by  using  a  fine  wire  plumb-bob,  the  angle  <p  could 
even  be  roughly  measured  by  a  Fraunhofer  micrometer,  showing  the  distance 
between  plumb-line  suspended  from  the  point  of  the  upper  pivot  and  point  of 
the  lower  pivot. 

A  "single-plate"  film  grating  (see  §60)  was  mounted  at  the  apex  of  the 

pendulum  triangle.    The  interference  rings  were  quite  strong  and  clear  and 

found  without  difficulty.   At  the  outset  it  is  possible  that  some  yield  of  metallic 

parts  may  be  registered,  though  the  yield  of  the  aluminum  tubing,  being  in 

)f  the  pendulum,  should  not  affect  its  reading  appreciably. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     47 


The  results  with  the  aluminum  pendulum  are  constructed  in  fig.  27,  the 
curves  showing  the  variation  of  inclination  in  seconds  of  arc  in  the  lapse  of 
time,  here  a  =  8.6  X  io~30  radian.  Since  8 = AAT/2  R = AAT/2 2 2 , 

a  =  3  -9  X  io~6  radian  =  8AAT  seconds  of  arc, 

the  present  factor  8  replacing  the  above  value  10.  As  in  the  case  of  §17, 
Part  I,  above,  the  present  results  are  intended  to  test  the  variations  from  fall 
to  winter  conditions,  and  during  the  introduction  of  steam  heat  into  the  labo- 
ratory. The  temperature  observations  are  therefore  also  inserted  in  the  dia- 
gram, so  far  as  necessary.  No  difficulty  whatever  was  experienced  with  the 
film  grating  throughout  the  whole  of  the  work. 


/^ 


a\ 


-X? 


?r~i 


T3' 


v 


2  c/fcr  / 


FIG.  27. 


In  the  first  half  of  the  observations  (upper  curve),  between  August  17  and 
September  28,  the  curve  shows  a  persistent  upward  trend.  Gaps  occur  in  the 
curve  at  a  and  b,  owing  to  the  absence  of  the  observer,  and  these  places  happen 
to  be  associated  with  variations  in  the  curve;  but  this  is  purely  incidental. 
The  curve  fails  to  get  back  to  its  original  reading.  It  seems  probable,  there- 
fore, that  the  cause  of  this  uniformly  progressive  march  is  the  viscosity  of  the 
aluminum  tubing  out  of  which  the  pendulum  was  built,  and  what  is  observed 
is  in  the  main  a  continuous  viscous  yield  of  the  pendulum  to  the  load  of  its 
own  weight. 

The  lower  curve  shows  the  results  between  September  28,  when  the  steam 
heat  was  turned  on,  and  November  4.  The  effect  of  the  sudden  appearance 
of  steam  heat  is  sufficiently  startling,  as  the  curve  on  October  i  runs  off  of 
the  scale  at  c.  It  was  then  necessary  to  displace  the  micrometer;  but  this 
was  done  so  as  to  change  the  fiducial  zero  as  little  as  possible.  Afterwards, 
however,  contrary  to  expectations,  the  curve  again  approaches  its  old  value, 
so  that  the  displacement  would  not  really  have  been  necessary.  What  took 


48     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

place  was  probably  something  like  this:  the  access  of  heat  in  the  room 
reaches  the  outer  layers  of  the  pier  first  and  only  gradually  penetrates  to  the 
interior  parts  of  the  masonry.  It  has  seemed  to  me,  also,  that  the  occurrence 
is  not  merely  a  question  of  temperature,  but  rather  a  case  of  drying  out  the 
parts  of  the  brickwork,  from  the  very  damp  conditions  in  the  summer  to  the 
desiccated  condition  during  the  winter  months.  The  lower  curve  is  naturally 
more  sinuous  than  the  upper,  but  not  nearly  as  much  so  as  would  have  been 
expected  in  comparison  with  the  curve  of  fig.  14  in  §  1 7 .  In  fact,  on  comparing 
these  two  curves  after  September  28,  the  initial  maximum  on  October  i  and 
the  minimum  on  October  6  correspond.  After  this  the  curves  diverge  and  there 
is  no  correspondence  until  the  maximum  of  November  i  is  reached.  Thus, 
for  instance,  on  October  19,  fig.  14  finds  no  counterpart  in  fig.  9.  A  compar- 
ison of  the  two  curves  is  naturally  immensely  in  favor  of  the  pier,  though 
even  the  latter  is  again  altogether  inadequate  for  the  kind  of  work  contem- 
plated; i.e.,  work  involving  variations  in  a  of  hundredths  of  second  of  arc. 
Changes  of  a  of  half  a  second  are  by  no  means  uncommon,  and  even  in  the 
summer  changes  of  2  seconds  within  a  month  would  be  a  small  estimate. 
The  curve  shows  the  nature  of  the  difficulties  encountered,  for  instance,  in 
endeavoring  to  measure  the  repulsion  of  two  disks  in  Chapter  II. 

Between  September  1 7  and  September  28  the  temperature  curve  (light  line) 
is  drawn  on  a  large  scale,  between  September  28  and  November  5  on  a  smaller 
scale.  With  regard  to  the  former  it  is  evident  that  the  sinuosity  of  the  curves 
is  about  the  same,  but  that  the  minima  and  maxima  are  not  cotemporaneous. 
Thus,  for  instance,  at  a  the  temperature  minimum  precedes  the  inclination 
minimum.  The  same  is  true  for  the  maximum  at  6,  etc.  In  their  details  and, 
in  general,  quantitatively,  the  two  curves  do  not  coincide  in  character.  Hence, 
the  effect  of  temperature,  if  admitted,  can  at  best  be  indirect;  i.e.,  tempera- 
ture changes  the  inclination  a  by  straining  or  warping  the  pier. 

If  we  compare  the  temperature  curves  between  September  28  and  November 
5  with  the  inclination  curve,  there  is  again  a  general  resemblance.  Thus  the 
maxima  at  /,  g,  k  and  the  minimum  at  h  occur  in  both.  But  there  is  no  detailed 
resemblance,  even  when  the  difference  of  scale  is  taken  into  consideration. 
Neither  is  the  temperature  effect  as  marked  as  in  the  corresponding  case  of 
fig.  14,  Part  I.  The  temperature  maxima  tend  to  precede  the  inclination 
maxima,  etc.  Hence,  as  before,  temperature  acts,  not  upon  the  pendulum 
mechanism  directly,  but  rather  indirectly  through  the  supports,  which  become 
displaced  by  unequal  expansions  in  the  pier  and  a  corresponding  tilting 
from  its  position. 

Finally,  the  changes  of  inclination  a  shown  by  the  aluminum  pendulum  are 
quite  as  marked  as  those  occurring  in  the  corresponding  case  of  the  steel  pen- 
dulum, although  the  viscosity  error  of  the  former  is  much  greater.  It  does 
not  therefore  appear  that  the  effect  of  changes  of  magnetic  field  has  produced 
any  error,  such  as  was  surmised  above  in  case  of  the  steel  pendulum.  The 
latter  is,  therefore,  preferable  for  work  of  the  present  kind. 


CHAPTER  II. 


THE  REPULSION  OF  TWO  METALLIC  DISKS,  NEARLY  IN  CONTACT. 

26.  Apparatus. — The  apparatus  shown  in  fig.  28  was  originally  constructed 
with  the  expectation  of  testing  the  horizontal  pendulum  for  the  measurement 
of  the  Newtonian  constant;  or,  conversely,  to  graduate  the  horizontal  pendu- 
lum by  means  of  that  constant.  Here  AB  suggests  the  parts  of  a  Fraunhofer 
slide  micrometer,  capable  of  moving  the  slide  about  6  cm.  and  graduated  in 
o.oooi  cm.  On  this  the  two  brass  disks  DD,  (originally)  15  cm.  in  diameter 
and  about  0.6  cm.  thick,  are  mounted  in  parallel,  rigidly,  normally  and  ver- 
tically. To  adjust  the  disks  the  steel  plugs  c  and  c  are  provided,  fitting  radial 
holes  in  the  plate.  They  are  further  held  by  the  semicircular  frame  e  and  e, 
screwed  to  the  slide  below  and  attached  above  to  the  disks  by  aid  of  the  pairs 
of  screws,  a  and  b,  on  opposite  sides  of  the  diameter.  The  screw  a  is  sunk  into 
the  disk,  while  6  presses  against  its  outer  surface.  As  the  disks  are  to  be  fitted 
nearly  true  to  the  slide  and  the  frame,  but  slight  adjustment  at  a  and  b  is  needed. 


1 

c 

d 

C.-4-: 

i- 

n 

|T 

c 

e> 

\\\ 

FIG.  28. 


FIG.  29. 


The  interior  of  the  smaller  disk  d,  (originally)  about  10  cm.  in  diameter  and 
0.6  cm.  thick,  is  suspended  vertically  by  two  fine  wires  /  from  the  end  of  the 
arm  of  the  horizontal  pendulum,  just  below  the  grating.  The  disks  D,  d,  D 
are  coaxial,  while  d  is  relatively  stationary;  D  or  D  may  be  brought  as  near 
to  d  as  desirable  by  aid  of  the  slide  micrometer,  the  other  disk  being  removed 
at  the  same  time. 


50     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

The  method  of  attachment  of  the  disk  d  to  the  horizontal  pendulum  is 
shown  on  a  smaller  scale  in  fig.  29.  Here  G  is  the  grating,  secured  by  three 
adjustment  screws  to  the  table  T,  the  cylindrical  shaft  of  which  is  grasped 
on  a  clamp  (open  form)  of  the  horizontal  pendulum  P.  To  the  bottom  of 
the  shaft  in  question,  a  cross-piece  hgh  is  screwed  and  fastened  with  a  lock-nut. 
The  two  fibers//  which  support  the  disk  d  are  wound  above  around  the  pulley 
screws  hh  and  thus  adequate  vertical  adjustment  of  disk  d  is  available. 

The  slide  micrometer  is  attached  to  the  pier  by  a  firm  horizontal  rail  capable 
of  adjustment  forward  and  rearward.  A  strong  clamp  attaches  the  base  of 
the  slide  micrometer  to  this  rail,  so  that  the  whole  instrument  may  also  be 
adjusted  to  the  right  or  left,  roughly.  The  fine  adjustment  is  completed  on 
the  slide  micrometer  itself. 

Finally  a  case  is  provided  covering  the  disks  D  and  d  and  part  of  the  micro- 
meter, so  that  only  the  drumhead  and  scale  projects.  The  apparatus  was  found 
to  work  satisfactorily.  It  is  quite  possible  to  reject  the  water  damper  at  the 
end  of  the  horizontal  pendulum,  above,  and  to  rely  solely  on  the  effective  air 
damping  produced,  when  the  disk  d  is  very  close  to  D  or  D'.  In  fact,  the  tin 
cover,  in  this  case,  was  all  but  superfluous.  D  could  be  shifted  from  end  to 
end  of  the  course,  without  materially  interfering  with  the  visibility  of  the 
ellipses  in  the  spectrum  of  the  interferometer.  The  real  interferences  unfavor- 
able to  the  gravitational  measurement  were  incidental,  due  either  to  the  change 
in  inclination  of  the  pier,  or  to  changes  in  the  magnetic  field  (inasmuch  as  the 
pendulum  was  preliminarily  constructed  of  steel  tubing),  or  to  the  causes 
discussed  in  this  chapter;  for  what  was  found  was  not  an  attraction  at  all, 
but  a  repulsion,  much  larger  in  absolute  value  than  the  attraction  anticipated. 

27.  Equations.—  The  chief  equations  to  be  used  in  the  present  work  have 
already  been  given  above.  It  is  merely  necessary  to  add  those  which  bear 
upon  the  sensitiveness  of  the  method.  Since  the  disk  of  mass  m  is  added,  at 
the  mean  distance  R,  to  the  mass  of  the  pendulum  M,  the  force  at  R  from  the 
axis  is  now 


The  gravitational  attraction  /'  of  the  disks  necessarily  involves  spherical 
harmonics,  but  may  be  written  temporarily  as 


where  m'  is  the  mass  of  the  stationary  disk  at  a  mean  distance  d  from  m. 
Equating  these  forces  and  inserting  the  value  of  FR,  the  equation  for  &N,  the 
displacement  at  the  micrometer,  becomes 


(3) 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     51 


In  the  first  place,  therefore, 


m/M 


m 

^f 

M 


so  that  the  lightest  available  pendulum  and  the  heaviest  admissible  disk  is 
to  be  selected,  although  the  increase  of  sensitiveness  is  not  quite  proportional 
to  m/M,  but  diminishes  as  (i+mR/Mh)-*.  This  procedure,  even  when  the 
float  is  used,  is  relatively  inefficient  and  the  value  of  AW  can  probably  not  be 
increased  more  than  twice  the  above  value  (a  difference  of  AW=  2X0.001  2 
for  the  two  extreme  positions  of  the  disk)  by  this  means. 

28.  Equations  for  the  vertical  pendulum.—  A  final  word  may  be  added  with 
regard  to  the  inclination  a.  This  can  be 
detected  with  such  precision  that  a  method 
based  upon  it  deserves  consideration. 
The  apparatus  in  this  case  would  take 
the  form  of  fig.  30,  where  ABCD  is  the  iron 
framework  of  the  heavy,  long,  vertical 
pendulum,  with  the  massive  bob  at  D  and 
knife-edges  and  tablets  at  et  so  that  the 
pendulum  is  capable  of  swinging  normally 
to  the  plane  of  the  diagram.  The  hori- 
zontal pendulum  is  attached  by  two  pivots,  f  cj) 

a  and  6,  to  the  central  rod  CD  of  the  ver- 
tical pendulum.    It  is  to  swing  clear  of  it  FlG>  3°- 

and  to  be  in  equilibrium  in  a  parallel  plane.  The  deflection  of  the  horizontal 
pendulum  is  also  normal  to  the  plane  of  the  diagram,  and  it  measures  the 
change  of  a  of  CD,  as  above,  G  being  the  grating,  h  the  center  of  gravity. 

When  gravitational  attraction  is  to  be  observed,  the  bob  D  is  one  of  the 
attracting  bodies  and  of  mass  m',  whereas  the  attracting  mass  m,  with  its 
center  on  the  same  level,  is  placed  in  front  of  or  behind  the  plane  of  the  diagram. 

If  the  mass  m'  at  the  end  of  the  vertical  pendulum  is  at  the  distance  L  from 
the  horizontal  axis,  and  the  mass  M'  of  the  remainder  of  the  pendulum 
virtually  at  a  distance  H  (center  of  gravity)  from  the  axis, 


where  d  is  the  mean  distance  apart  of  m  and  m'.    Hence 

(6)  AW'  =  ym2R/<pd2g(i  +M'H/m'L) 

If  m'  is  massive,  so  that  M'H/m'L  =  i  may  be  assumed;  if  the  bodies  m  and 

m'  are  equal  spheres  of  radius  r  all  but  in  contact, 

d=2r  and  &N'  =  yirpRd/6<(>g 

Thus  if  d=io  cm.,  p=io,  with  the  other  magnitudes  as  in  the  above  inter- 
ferometer, AW'  =  io-6X7.5  cm.,  which  increases  but  as  the  first  power  of  the 
diameter  of  the  spheres.  Hence,  in  spite  of  the  precision  of  a  measurement, 
the  method  would  not  be  available  for  the  determination  of  7. 


52     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

29.  Observations  with  small  plates.— The  first  experiments  were  made 
merely  for  the  purpose  of  testing  the  method,  using  the  same  heavy  horizontal 
pendulum  as  in  the  preceding  section.  There  are  two  or  three  objections  to 
this  pendulum  for  the  present  purposes,  of  which  the  first  is  its  weight  M;  the 
second  is  the  water  damper,  which  introduces  inevitable  discrepancies,  due  to 
such  capillary  forces  as  result  from  surface  viscosity.  The  third  objection  is 
due  to  the  fact  that  the  pendulum  is  made  of  light  steel  tubing  and  points  in 
the  north-south  direction.  These  tubes  become  weak  magnets  in  the  earth's 
field,  and  the  angle  6  may  change  with  the  variations  of  this  field.  Finally  the 
inclination  a  of  the  axis  of  the  pendulum,  due  to  terrestrial  causes,  is  itself  to 
be  considered;  this  can  only  be  eliminated  if  the  time  of  observation  is  reduced. 

The  two  attracting  plates  of  rolled  brass  were  each  6  inches  in  diameter 
and  0.25  inch  thick,  weighing  w'  =  i,o3S  grams.  The  attracted  disk  d  at- 
tached to  the  horizontal  pendulum  was  4  inches  in  diameter  and  0.125  inch 
thick,  weighing  227  grams.  The  distance  between  the  large  plates  was  2 . 5  cm. 
on  the  micrometer,  this  being  about  the  limit  of  the  micrometer  screw  and 
sufficient  for  the  diminution  of  the  attraction  in  question  to  negligible  values. 
The  difference  of  AAT  for  the  two  extreme  positions  of  the  disks  was  esti- 
mated above  as  0.0024  cm.,  or  5  drum-parts.  It  should  have  been  easily 
detected,  if  not  masked  by  the  incidental  disturbances  referred  to. 

The  five  series  of  observations  are  given  in  the  curves,  figs.  3iA,  316,  31 C, 
32A  and  326.  They  show  both  the  release  of  the  suspended  disks  from  con- 
tact with  the  disk  fixed  on  the  micrometer,  and  the  differential  effect  of  the 
fixed  disks  on  opposite  sides  of  the  suspended  disk,  but  near  it. 


Z\fl 


A 


0     40     -20    30    40  -05     &     & 

FIG.  31. 

In  fig.  3 1  A,  the  abscissas  are  the  successive  excursions  A*  of  the  micrometer 
bearing  the  fixed  plates,  the  ordinates  are  the  corresponding  excursions  AN 
of  the  suspended  plate.  Beginning  at  a,  the  two  plates  are  nearly  in  contact, 
and  this  contact  is  made  more  definite  in  the  direction  +  x.  Hence  in  the 
curve  from  a  to  d  to  6,  as  shown  by  the  arrows,  2A/V=A*,  as  it  should  be. 
After  passing  b  toward  c  the  suspended  plate  is  released,  but  released  in  such 
a  way  as  to  suggest  repulsion  at  6,  whereas  the  other  four  points  nearer  c 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      53 


in  their  downward  slope  toward  the  left  would  be  compatible  with  gravita- 
tion. Release  should  take  place  at  the  intersection  of  the  two  lines.  Results 
of  the  same  kind  are  shown  in  fig.  3iB,  where  the  apparent  repulsion  is  very 
definite.  It  is  probable  that  in  both  cases  the  discrepancies  observed  are  dis- 
torted by  capillary  forces,  surface  viscosity  at  the  water  damper,  and  by  the 
inclination  difficulties.  This  is  borne  out  by  fig.  3  iC,  in  which  the  fixed  disks 
were  alternately  placed  all  but  in  contact  with  the  suspended  disk.  The  curve 
should  have  been  zigzag,  with  the  oscillations  equal  and  in  opposite  directions; 
but  it  is  quite  irregular,  due  to  extraneous  causes.  R  and  L  indicate  whether 
the  fixed  disk  is  on  the  right  or  left  side  of  the  suspended  disk. 
A  v  B 


•A 
•3 
•t 

4 

ft 

\ 

\ 

X 

•005 

f 

4 

\ 

•COO 
-005 

w* 

T 

i 

\ 

-a 


FIG.  32. 


The  water  damper  was  now  removed  and  the  work  repeated,  relying  on 
the  air-damping  at  the  disks  only.  No  difficulty  was  experienced  in  obtaining 
the  interferences;  but  the  results  fig.  32A  show  no  evidence  whatever  of  attrac- 
tion. Similarly  in  the  alternations  of  fig.  326,  the  curve  which  should  have 
been  zigzag  shows  no  regularity.  Here  again  foreign  disturbances  have 
masked  the  effect  sought,  although  the  displacements  themselves  were  ap- 
parently definite  and  satisfactory.  It  is  therefore  necessary  to  replace  the 
disks  by  a  larger  set,  as  is  done  in  the  next  section. 

30.  Observations.  Plates  of  larger  area. — The  brass  plates  were  now  re- 
placed by  a  set  larger  in  area  but  thinner,  this  being  in  the  direction  of  the 
improvement  of  method  indicated.  The  same  unnecessarily  heavy  steel 
pendulum  had,  however,  to  be  used,  so  that  M=i,2so  grams,  h  =  So  cm., 
R  =  111.3  cm.,  ^  =  0.01081  radian,  ^  =  42.9  AAf.  The  new  brass  plates  were 
identical  in  size,  the  mass  being  m  =  468  grams  each,  the  diameter  zr  =  20.3  cm., 
and  the  thickness  0.17  cm.  In  place  of  gravitational  attraction  an  apparent 
repulsion,  equivalent  on  the  average  to  0.0338  cm.,  or  about  68  drum-parts, 
was  observed. 

The  observations  are  given  in  table  i  and  in  figs.  33A,  336,  330,  the  arrows 
showing  the  direction  of  successive  observations.  The  abscissas  denote  the 
positions  A*  of  the  attracting  "fixed  "  plate  on  the  micrometer,  the  ordinates 
the  corresponding  value  of  the  displacement  AN  of  the  plate  suspended  from 


54     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

the  horizontal  pendulum,  read  off  on  the  micrometer  of  the  interferometer. 
The  plan  was  to  begin  with  the  plates  more  than  in  contact,  so  that  the  mov- 
able disk  is  carried  by  the  fixed  disks  until  released.  In  fig.  3aA,  at  b,  the 
plates  adhere  until  at  a  a  release  or  fall  suddenly  takes  place,  the  plates  being 
now  over  i  mm.  apart.  The  figure  shows  that  release  should  have ^ occurred 
at  the  position  2.09  cm.  In  fig.  33B,  corresponding  to  the  other  ("small") 
side  of  the  Fraunhofer  carriage,  the  plate  is  released  at  the  position  0.51  cm. 
and  there  is  no  adhesion.  In  fig.  33 C,  on  the  original  ("large ")  side,  the  plate 
is  passed  into  cohesion  and  then  released  with  a  smaller  fall  at  a. 

TABLE  I  — Large  brass  disks,  not  in  metallic  contact.  Steel  horizontal  pendulum,  m  =468 
grams;  r  =  10.2  cm.;  *=o.i7  cm.;  ^  =  0.0108;  M=  1,250 grams;  h  =  80 cm.;  £  =  111.3  cm.; 
F  = 


Fixed  plate  at 
Ax 

Movable  plate  at 
io*AN 

Fixed  plate  at 
A* 

Movable  plate  at 
io4AW 

Fig.  33A.  2.15  cm. 

.10 

•05 

.00 

1-95 
.90 

-  955 
+    37 
1030 
2015 
2875 
283 

Loosening.  2.05  cm. 
2.05 

2.00 
2.05 

+650 
185 
295 
215 

Fig.  338.  0.45 
.40 
•50 

:io 

•65 

2493 
3493 
1605 
625 
495 
495 

Fig.33D.  2.00 
/.6o 

Us 

2.00 

•65 

2.00 

•65 
2.0O 

•65 

•302 
580 
517 
187 
565 
ISO 
535 
177 
550 

Fig.33C.  0.65 
1.90 
1-95 

2.00 
2.05 
2.10 

495 
285 
320 
320 
+  175 
-  285 

*  Mean  values. 

Thus  the  positions  2.00  and  0.65  are  guaranteed  as  free,  the  space  between 
the  reacting  plates  being  over  i  mm.,  as  compared  with  the  distance  1.4  cm. 
between  the  fixed  plates.  The  effects  of  alternately  approaching  the  opposed 
fixed  plates  to  the  movable  disk  are  shown  in  fig.  330.  They  are  quite  definite, 
larger  in  order  of  value  than  would  be  anticipated  and  constitute  repulsions 
instead  of  attractions.  In  fact,  figs.  33A  and  336  show  that  in  case  of  attrac- 
tion or  of  cohesion,  AJV  should  be  too  large  on  the  "large  "  side,  and  too  small 
on  the  "  small "  side  of  the  stationary  disk.  In  fig.  330  the  reverse  is  the  case. 

To  explain  this  repulsion  a  number  of  facts  have  to  be  taken  into  ac- 
count. Both  the  fixed  disks  are  separate  metallic  systems,  but  ultimately 
anchored  into  the  pier  with  iron  bolts,  so  that  a  volta  contact  force,  iron-brass, 
would  be  inevitable.  The  disks  are  thus  carrying  charges,  depending  on  the 
nature  of  the  anchorage  in  the  pier,  whether  this  is  moist  or  quite  dry.  It  seems 
probable,  as  will  be  shown  below,  that  these  small  potentials  are  negligible. 
Again,  with  the  small  forces  per  square  centimeter  of  area  in  question,  the 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     55 


viscosity  of  air  is  sufficient  to  necessitate  the  lapse  of  considerable  time  before 
a  position  of  equilibrium  is  assured,  even  with  the  disks  a  millimeter  or  more 
apart.  In  the  above  experiment  sufficient  time  was  allowed  until  the  motion 
became  vibratory,  after  which  the  reading  was  taken;  but  it  is  difficult  to 
assert  that,  even  after  indefinite  waiting,  further  subsidence  would  not  have 
taken  place.  Finally  the  flexure  or  tipping  of  the  pier,  where  long  intervals 
of  time  are  in  question,  can  not  be  eliminated.  There  would  inevitably  be 
some  error  on  this  account.  It  seems  improbable,  therefore,  that  the  actual 
gravitational  attraction  of  metallic  disks  will  be  determinable,  while  a  non- 
metallic  system  is  liable  to  introduce  even  greater  errors. 

D 


\ 


ftO 


20     fcf    4 


-5-6 

D 

FIG.  33. 


1-95 


\^, 


31.  The  same,  continued.  Metallic  contact.  —  The  next  advance  consisted 
in  placing  the  disks  in  electrical  (metallic)  contact,  which  was  easily  done  by 
joining  the  pivots  of  the  horizontal  pendulum  with  the  slide  of  the  micrometer 
bearing  the  fixed  disks  by  a  copper  wire.  Moreover,  since  the  position  of 
equilibrium  is  gradually  reached  in  the  lapse  of  minutes,  the  time  of  the 
observations  is  taken  in  minutes.  These  results  are  given  in  table  2,  and  are 
inscribed  in  figs.  34A  and  346.  The  figures  on  the  curve  show  the  series  in 
question  and  the  plate  ("large"  or  "small"  side  of  the  plate  micrometer), 
which  is  actively  repelling.  In  fig.  34A  the  alternations  are  found  after  long- 
waiting;  in  fig.  343,  however,  in  time  series.  When  equilibrium  is  reached, 
and  this  is  always  relatively  quickly,  the  disk  oscillates  due  to  incidental 
causes.  It  makes  no  difference  from  which  side  the  position  of  equilibrium 
is  approached  (series  2,  8,  15).  The  presence  of  radium  on  the  plates  has  no 
effect,  other  than  the  mechanical  disturbance  given  by  placing  it  there;  the 
same  position  of  equilibrium  again  results.  When  the  disks  are  jolted  by 
contact  (case  between  series  7  and  8),  the  equilibrium  position  may  be  tem- 


56     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


•iO 


10  a 


•06 


15 


04- 


•02 


•00 


-OS, 


/\ 

10 


B 
FIG.  34. 


50T 


TABLE  2. — Large  brass  disks  in  metallic  contact.    Constants  as  in  table  I. 


Ax 

lO'AW 

Time. 

Ax 

IO*Atf 

Time. 

Ax 

itfAAT 

Time. 

cm. 
0.65 

2.00 

•65 

2.00 
2.05 
2.10 

cm. 

3 

43 
29 
29 
26 

min. 

cm. 

Vi5 

cm. 
68 
64 

68 

min. 

0 

2 

4 

cm. 

2£ 

cm. 

% 

49 

45 

min. 

0 

i 

2 

4 

VI? 

28 
49 
55 
*50 
53 
53 

0 

i 

2 

4 
5 
6 

°& 

90 
68 
63 
60 

0 

2 

4 

2.20 
I 

0 

29 
36 
4i 
46 
46 
47 

0 
2 

4 
6 
8 

10 
12 

0-75 
XIII 
Radium  on 

69 
65 
62 
62 

6 

7 
8 
9 

0.65 

144 

0-75 
VIII 

-  7 

21 

tS3 

*62 

66 

0 

3 
5 

7 

2i'i5 

132 
103 
t-50 

0 
2 
10 

xft 

Radium  off 

62 
62 

12 
14 

0.80 
III 

7i 
68 
?59 
70 

O 
2 

4 
6 

2rJ? 

36 
50 

*48 
48 
48 
48 

o 

2 

3 
4 
5 
53 

xv5 

30 
48 
50 
52 

0 

I 
2 

4 

°# 

66 
67 
65 

o 

2 

4 

2.15 

Radium  on 

59 

54 

T 

88 
68 
60 
61 
60 

0 

i 

2 

4 
6 

'$' 

48 
56 
55 

0 
2 

4 

Vibrating. 


t  Fall. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     57 


porarily  disturbed  (series  8).  Apart  from  flexure  of  the  pier,  long  waiting 
(50  minutes  in  series  9)  does  not  further  change  the  position  of  equilibrium,  if 
the  slight  swinging  is  taken  into  account.  The  cause  of  the  gradual  motion 
may  reside  in  the  viscosity  of  air,  as  indicated  in  the  next  paragraph. 

If  we  compare  the  results  of  fig.  330  (system  not  in  metallic  contact)  with 
the  present  (system  metallically  connected)  the  results  appear  as  follows: 


System  not  in 
metallic  contact 

System  metallically 
connected. 

2  AW  =  0.02  1 
22 

2AJV=o.oi6 
H 

38 
38 

37 

•015 

12 
13 
10 
II 

Mean  2  &N=  0.038 

Mean  2AJV  =0.013 

The  disks  were  usually  about  a  millimeter  apart.  Metallic  contact  has 
thus  apparently  made  the  repulsion  smaller;  but  it  is  not  certain  that  the 
distance  apart  of  the  plates  is  quite  identical.  Moreover,  data  obtained  at 
different  times  vary  considerably.  In  the  present  case  the  repulsion  observed 
for  the  disks  20  cm.  in  diameter  is  2^  =  65.2^=65. 2X0. 013  =0.85  dyne, 
at  about  d  =  i  mm.  of  air-space. 

32.  Retardation  due  to  viscosity  of  air. — It  will  next  be  necessary  to  ex- 
amine the  above  suggestion,  that  the  very  gradual  approach  of  the  suspended 
disk  to  its  position  of  equilibrium  may  be  due  to  the  viscosity  of  the  interposed 
film  of  air,  in  view  of  the  small  forces  and  small  displacements  involved. 
The  case  may  perhaps  be  treated  in  terms  of  Poiseuille's  law,  assuming  that 
the  flow  is  from  the  center  of  the  two  nearly  contiguous  parallel  disks  radially 
toward  the  circumference.  Let  y0  be  the  initial  distance  apart  of  the  disks, 
and  the  time  t  =  o  second,  measured  from  the  fixed  toward  the  movable  disk. 
Let  y'  be  the  final  position  of  equilibrium  of  the  movable  disk,  so  that  its 
excursion  is  y0— y'.  Let  a  small  impulsive  force  P  act  normally  on  the  outside 
of  the  movable  disk,  by  which  it  is  put  into  the  position  y.  The  pressure 
generated  will  cause  a  flow  radially  outward,  and  if  p  is  the  pressure  in  the 
fluid  at  a  distance  r  from  the  center,  Poiseuille's  law  may  be  written 

d)  -*v=    fr=Cwy)'<fr 

dt  ~  8irJj       dr 

for  the  flow  through  a  ring  whose  section  is  y.dr,  if  i  is  the  viscosity  of  the  gas 
and  V  the  volume  of  fluid  crossing  per  second.  If  the  flow  is  steady,  so  that 
dp/dt  =  o  for  all  distances  from  the  center,  and  if  the  liquid  is  virtually 
incompressible,  i.e.,  V  independent  of  r,  the  problem  may  be  solved  without 
difficulty.  Neither  of  these  conditions  is  quite  true.  The  second,  however, 
inasmuch  as  the  average  pressure  increment  is  exceedingly  small  relative 


v= _ 

V         dt 


58     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

to  atmospheric  pressure,  may  be  admitted.  Suppose,  therefore,  that  at 
any  time,  y  and  V  are  constant  relatively  to  r,  and  integrate  the  equation. 
Then  the  pressure  excess  at  r  is 

If  R  is  the  radius  of  the  disks,  p  =  o  at  r=R,  and  the  equation  may  be  con- 
sidered to  hold  short  of  r=o.  Thus  the  thrust  P  becomes 


(3) 


But 

(4) 
whence 

(5) 

or,  on  second  integration, 

(6) 

But  for  the  horizontal  pendulum  the  force  P  is  proportional  to  y—yr,  which 
may  be  written 

so  that  (5)  becomes 

(8)  PtfU = 2vt)P?dy/yl(y—y') 

This,  on  again  integrating,  becomes  finally 

(9) 

natural  logarithms  being  in  question.  Since  y—y'=bN/2,  equation  (7)  cor- 
responds in  case  of  the  large  disks  to  P=FR  =  2X65.2&N/2.  Hence,  PO  = 
2X65.2.  Furthermore,  in  case  of  this  apparatus  and  in  the  present  experi- 
ments, the  following  data  may  be  entered :  P0  =  2  X  6  5 . 2 ;  y'  =  i  / 1 5  or  y'/y0  = 
2/3;  #=10.2  cm.;  j>0  =  0.10  cm.;  >j  =  i9oXio-«,  so  that  roughly 


Table  3  contains  some  corresponding  values  of  y  and  P  computed  in  this  way. 
TABLE  3.— Motion  of  movable  brass  disk  retarded  by  viscosity  of  air  film. 


io4X 

y=AN/2 

> 

io*  X 

t 

cm. 

sec. 

cm. 

sec. 

IOOO 

0.00 

680 

548 

900 
850 

•39 
.70 

670 
668 

8.47 
10.47 

800 

1.16 

666.8 

19.41 

750 

i-93 

y'  =  1/15 

oo 

700 

3-68 

EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      59 

These  values  are  reproduced  in  fig.  35,  which  with  table  3  shows  that  the 
position  of  equilibrium  is  reached  to  AN=io~4,  the  smallest  quantity  easily 
measurable,  in  about  25  seconds. 


8      10 
FIG.  35. 


/4-      1  6 


The  fluid  has  been  treated  as  incompressible.  If  this  is  not  done,  the  results 
apparently  become  unavailable.  A  further  step  may,  however,  be  made: 
Poiseuille's  equation  (i)  if  the  condition  VP  =  Vo  PO  is  introduced,  leads  on 
integration  to  the  form 

(n)  P'-p'o 

where  P  is  the  pressure  and  Vo  the  volume  issuing  at  the  edge,  per  second  at 
the  normal  pressure  PQ.  In  endeavoring  to  use  (n)  directly,  I  have  not 
succeeded  in  producing  a  practical  form  of  equation. 

Equation  (9)  may  be  put  in  a  different  form  suitable  for  computing  in  the 
ultimate  times  of  very  close  approach  to  equilibrium.  For  this  purpose,  let 

y'/yo  =  a  and  y—y'=b 

where  6  is  to  be  very  small,  so  that  y=ayo+b.  Equation  (9)  then  reduces 
nearly  to 


ay 

Usually  b/ay0  may  be  neglected  compared  with  i—  a.  Thus  if  6  =  io~4  cm., 
t  =  1 1 .  i  sec. ,  with  the  other  constants  as  above,  y0 = o.  i  cm. ;  a  =  2/3 .  For  the 
same  case,  6  =  io~4  cm.,  if  70  =  0.05  cm.,  0  =  2/3,  ^  =  38.3  sec.,  are  needed  to 
approach  within  io~4  cm.  of  the  position  of  equilibrium,  etc.  In  case  of  repul- 
sion, a>  i  and  6  is  negative.  Thus  for  0=3/2  cm.,  6=  io~4  cm.,  yQ=  1/15  cm., 
*  =  6.53  sec.  For  70  =  2/45  cm.,  y'  =  1/15  cm.,  /  =  13.8  sec.,  etc.  The  intervals 
so  computed  are  small  as  compared  with  the  times  actually  observed,  where 
many  minutes  have  to  elapse  before  equilibrium  is  obtained.  It  seems  diffi- 


60     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

cult  to  interpret  this  excess  by  supposing  that  the  method  is  inadequate;  for 
the  effect  of  gravitational  attraction  between  the  disks,  which  has  been 
ignored,  would  be  a  virtual  increase  of  P0.  Since,  on  the  average,  pressure 
excess,  p— po,  is  very  small  as  compared  with  pQ  in  the  actual  case,  equation 
(n),  seeing  that  p*-p*o  =  2p<>  (p-po)  becomes 


and 


which  is  identical  in  interpretation  with  equation  (2)  above,  where  p0  = 
therefore  leads  to  the  same  conclusion. 


33.  Observations,  continued.  Presence  and  absence  of  electrical  contact.— 

Notwithstanding  the  improbability  of  electrical  effects,  it  was  thought  neces- 
sary to  test  the  case  directly.  Accordingly,  in  table  4  and  fig.  36,  series  i 
to  8,  experiments  are  recorded  with  the  plates  not  in  metallic  contact,  series 
i  to  5,  and  with  the  plates  in  metallic  contact,  series  6  to  8,  respectively.  The 
behavior  in  both  cases  is  virtually  the  same,  when  the  shift  of  zero  is  taken 
into  account.  Observations  are  plotted  in  time  series,  with  the  last  observa- 
tion marked  by  a  circle,  and  they  are  in  each  case  continued  until  the  motion 
of  the  plate  is  retrograde,  whereupon  the  real  oscillation  of  the  plate  begins. 
To  throw  further  light  on  the  subject,  a  Leclanche'  cell  was  introduced  in 
series  9  and  10  and  removed  in  series  1  1.  The  lighting  circuit  of  the  room  was 
placed  more  remote  in  series  12  and  the  system  earthed  in  series  13  and  14 
for  both  fixed  disks. 


•iO 
08 
06 
04 
•02 
•00 

( 

1 

* 

r 

I 

5 
N* 

8 

b 

8 
I 

»?1 

'    a 
ft 

£ 

5 

10 

•70 

70 

•70 

I 

• 

/ 

10 

L 

« 

f 

2 

-» 

2/6 

1« 

r 

fs 

d 

/ 

r 

I 

C 

\ 

A 

t 

/a  y 

$ 

I 

& 

me,— 

*• 

'  K 

«  -  ty 

05 

2<M 

pp 

T*     1 

40       60       80      iOO      ffl      i40  1   160     1 

0     £6 

0     & 

0     2(0 

FIG.  36. 

That  the  differences  observed  are  most  probably  referable  to  the  flexure  of 
the  pier  in  shifting  the  zero,  is  shown  in  series  15  to  1 8,  where  the  observations 
are  made  on  one  side  only,  with  the  distance  between  disks  gradually  increas- 
ing, as  the  fixed  plate  moves  from  position  2.15  to  position  2.00,  i.e.,  0.15  cm. 
The  interval  of  observation  was  48  minutes;  but  the  interval  and  final  reading 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     61 

TABLE  4.— Displacement  of  brass  disk  with  electrical  contact  and  without.    Constants  as 

in  table  I. 


Ax 

itfAN 

Time 

Ax 

lo'A^V 

Time 

AJC 

i&AN 

Time 

A* 

itf&N 

Time 

cm. 
0.65 

cm. 
93 

mm. 
46 

cm. 
2-15 

cm. 
8 
27 
35 
39 
43 
49 
52 
52 
52 

min. 
46 
48 
50 
52 
54 
56 

62 

cm. 
0.70 
XI 

cm. 

1 

min. 
31 
33 
35 

cm. 

2.00 

XVII 

cm. 
55 
ii 
ii 
20 

min. 
46 
48 
50 
52 

0.70 

g 
| 

48 
50 
52 
54 

VII 

0.70 
XII 

Earthed 

83 

77 

8 

2.15 
XVIII 

0 

ii 

18 
18 

58 
60 
62 
64 

2i'i5 

-  6 

+22 
28 
38 

45 

44 

57 
59 
6l 

63 
65 
67 

2.15 

XIV 

—  12 

+25 

n 

35 

50 
52 

1 

0.70 
VIII 

104 
81 
69 

*83 
*73 

i 

8 

IO 
12 

A* 

itfAW 

Time 

cm. 
0-75 
XIX 

cm. 
36 
33 
33 
24 
15 

min. 
7 
9 

10 
12 
14 

0.70 

III 

98 

f7 

8 

IO 
12 
H 

16 

2.15 

2.IO 

XV 

45 

$ 

11 

26 
34 
23 

16 
18 
19 

21 
23 
25 
27 
31 

0.70 
IX 
Battery 
on 

76 

75 

75 

16 
18 

20 

2-15 
IV 

7 
26 
35 
46 
47 

19 

21 
23 
25 
27 

$ 

100 

67 
57 
50 
47 
47 
47 

•i|4 
'M 

17 
19 
21 

23 
25 
27 
29 

0.70 
X 

Poles  ex- 
changed 

I? 

83 

22 
24 

26 

T 

88 

II 

30 
32 

2.05 
XVI 

57 
32 
29 
25 
15 
29 

12 

33 
35 
37 
39 
41 
43 
45 

34 

0.70 

Battery 
off 

83 

28 

0.70 

VI 

69 

88 
86 

39 
41 
43 

*Next  day,  71. 

at  2.15  cm.  differ  in  like  degree,  so  that  the  apparent  attraction  indicated  is 
merely  the  result  of  the  shift  of  the  position  of  equilibrium  of  the  horizontal 
pendulum.  Series  19  and  20  contain  similar  observations  on  the  other  side. 

34.  Observations,  continued.  Change  of  distance  apart. — In  the  following 
work  table  5  and  figs.  37,  38,  3gA,  398,  the  attempt  is  again  made  to  vary 
the  distances  between  fixed  and  movable  plates,  successively,  but  to  determine 
the  micrometer  position  of  contact  of  the  plates  by  actually  pushing  them  to- 
gether with  a  weak  spring.  Observations  are  made  on  the  "larger"  side  in 
series  i  to  8,  the  distance  apart  increasing  from  position  2.15,  where  it  is 
0.045/2  cm.,  to  position  2.05,  where  it  is  0.247/2  cm.,  i.e.,  the  total  displace- 
ment being  about  o.i  cm.  The  observations  are  given  in  time  series  in  fig. 
37  and  are  not  interrupted,  until  the  motion  of  the  pendulum  is  retrograde. 
The  doubly  inflected  curve,  series  3,  4,  5,  etc.,  is  well  shown;  i.e.,  the  plates 
after  immediate  contact  separate  very  slowly,  whereupon  the  speed  of  sepa- 
ration reaches  a  maximum,  to  decrease  to  zero  again  when  the  pendulum 
regains  its  position  of  equilibrium.  It  is  probable  that  in  series  i  and  2  the 


62     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


20     40      60      80 
FIG.  37. 


140 


pendulum  has  not  separated  as  far  as  its  position  of  equilibrium,  the  motion 
here  being  excessively  slow.  The  whole  of  the  motion  in  series  i  to  8  may  be 
followed  by  moving  the  micrometer  as  the  ellipses  pass  through  the  spectrum. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     63 


If  the  mean  results  be  taken  from  the  figure,  the  following  data  appear: 
Position  of  fixed  plate 2.05    cm.        2.10    cm.        2.15    cm. 


AN.    Plates  in  contact 247 

AN.    Suspended  plate  free ooo 

AN.    Distance  apart  of  movable  and  fixed  plate  . . .     .247 


.165 
.010 
.165 


.067 
.022 
•045 


These  results  are  shown  in  fig.  3pA.  They  are  such  as  to  indicate  an  appa- 
rent attraction  of  the  fixed  and  movable  disks,  increasing  as  their  distance 
apart  diminishes.  Observations  were  now  made  on  the  other  side  to  see 
whether  the  results  would  be  corroborated,  or  whether  in  the  two  hours  of 
observation  the  shift  of  the  position  of  equilibrium  of  the  pendulum  was  so 
A  B 


0.65   cm. 
—  .050 
.064 


0.70   cm. 
-.142 
.070 

.212 


0.75    cm. 
—  .240 
.071 


great  as  to  obscure  the  true  conditions  completely.  The  new  results  for  the 
"  small  "  side  are  given  in  fig.  38.  They  exhibit  the  familiar  inflected  curves 
whenever  the  pendulum  separates  from  the  position  of  contact,  which  are 
observed  until  the  pendulum  begins  to  swing  to  and  fro.  In  series  14  and  15 
the  movable  disk  was  probably  not  quite  free.  The  mean  results  may  be 
estimated  as 

Position  of  fixed  plates 0.60    cm. 

AN.    Plates  in  contact 055 

AN.    Suspended  plate  free 086 

AN.    Distance  apart  of  fixed  and  mov- 
able plate 031  .114  .212  .311 

When  the  plates  are  0.150  cm.  apart  (0.75),  the  air-damping  is  insufficient. 
The  ellipses  move  to  and  fro  in  the  spectrum.  These  results  are  given  in  fig. 
396,  and,  apart  from  the  observation  at  0.60,  they  contain  no  evidence  of  any 
gravitational  effect  beyond  the  limits  of  error.  The  time  needed  was  2 
hours,  within  which  the  shift  of  the  position  of  equilibrium  of  the  horizontal 
pendulum  is  not  guaranteed. 

It  does  not  seem  possible,  therefore,  to  obtain  any  definite  results  from 
methods  which  necessarily  consume  as  much  time  as  the  present. 


64     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


TABLE  5. — Brass  disks  alternately  in  contact  and  free  on  one  side.    Constants  as  in  table  I . 


Remarks. 

Ax 

I03XAAT 

Time. 

Remarks. 

Ax 

lO'Atf 

Time. 

Free  
Contact  .  .  . 
Free  
I 

cm. 
2.15 
2-15 
2.15 

cm. 

21 

68 
37 
34 
33 
33 

min. 

53 
55 
57 
59 

Contact  
Free  

cm. 
2.05 
2.05 

cm. 
244 
226 
205 

_1 

+      2 

min. 

's 

10 
12 

;j 

18 

VIII 

Contact  .  .  . 

2.15 
2-15 

66 
36 

34 
33 
33 

3 
5 
7 
9 

Free  
II 

The  other  side,  next  day. 

Contact  .  .  . 
Free  
IX 

0-75 

•75 

-240 
—  200 

+  67 

i 

3 
5 

Free  
Ill 

2.IO 

126 
"3 
99 
60 
16 
8 
9 

12 
14 

16 
19 

22 

11 

Free  

Contact  .  .  . 
Free  
XI 

0.70 

0.70 
.70 

76 
65 
68 
-142 

8 

10 
12 

20 
22 
24 
26 

Contact  .  . 
Free  

2.10 
2.10 

164 
135 
131 
127 
119 
106 
84 
49 

20 
13 

9 

10 

31 

33 
35 
37 
39 
4i 
43 
45 
47 
49 
51 

215 
-  67 
+  44 
+  61 

IV 

Free... 
XII 

Contact  .  . 
Free  

0.65 

il 

87 

H 
68 
-  50 
—   II 

+  i 

39 
50 
62 
66 
68 
61 

29 
31 

33 

35 

47 
49 
51 
55 
57 
59 
6l 

63 

65 

XIII 

Contact  .  . 
Free  
V 

Free.. 

2.10 
2.IO 

2.10 

166 
124 

112 

99 

78 

-1 

+      2 
12 

56 
58 
60 
62 
64 
66 
68 
70 
3  hours 
later 

Free.  .  . 

0.60 

123 
104 
99 
95 

93 

87 
86 
86 

9 
II 

13 
15 
17 
19 

21 

24 

27 
30 

44 

XIV 

Free... 
VI 

2.05 

O 
*+      2 
-    17 

-     4 

42 
44 

jf 

Contact  

2.05 

250 

Free... 

2.05 

228 
208 

'1 

o 

8 

it 
8 

62 

64 

Contact  .... 
•p___ 

0.60 
.60 

II 

86 
86 

5i 
55 
67 

VII 

XlVf 

•Swinging.           fTotal  time  of  observation,  over  2  hours. 

EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     65 


35.  Observations.  Long  periods  and  inversion.—  The  unsatisfactory  results 
obtained  in  the  last  paragraph  induced  me  to  give  a  final  trial  to  the  original 
method  of  alternating  the 


sign  of  the  repulsion,   by 
moving   each    of  the   two     -/oIL  

I 

fixed  plates  in  turn  near          1 
the  suspended  plate.     The     -0ft  1 

I 

results  are  given  in  table 
6  and  fig.  40.     In  series  it     "06\\ 

4^ 

t 

2,  3  the  equilibrium  posi- 
tion   is    approached    from    -04 

\fi 

•70 

I   J 

M 

t 

65 

two  opposed  directions  and 
for  two   positions   on  the    -02    1  / 

!» 

|| 

large   side.     The  same   is             \ 
thp  rasp  iri  RAri^s  5  and  6      -00  M| 

L 

10 

215 

IH~H 

/H 

while  in  series  4,   7,  8,  9,         fS 
similar     observations     are  -021  

**v 

c|i 

e--* 

1     \ 

made   on   the  small    side.        ^    ^ 
The  mean  results  are 
Fixed  disk  at                               2.102  15 

0     4Q      60      80     100     120    m 
FIG.  40. 

0.70      2.15      0.75      0.70      0.65    cm. 
5  -.150      .075       150 
t      .045      .016      .053       .052       .048  cm. 

.007         .010                           .101 

AN.    Plates  in  contact  .07 

AN.    Movable  plate  free  0.015    .01: 

d=...                                                            .03] 

Hence  there  can  be  no  further  doubt  that  the  repulsions  are  real,  although 
their  nature  has  not  been  made  out.  When  the  distance  between  the  disks 
is  larger  than  a  millimeter,  the  air-damping  is  insufficient  and  the  free  disk 
unavoidably  oscillates,  as,  for  instance,  in  case  of  series  i  and  4.  The  evidence, 
however,  is  none  the  less  definite.  In  series  i  and  3,  5  and  6,  7  and  9,  the  equi- 
librium position  is  approached  from  opposite  directions  (the  displacements  of 
the  horizontal  pendulum  are  in  half  centimeters). 

In  order  to  obtain  some  reason  for  this  result,  one  may  dismiss  the  effect  of 
electrical  repulsion  at  once.  Experiments,  moreover,  are  to  be  made  in  the 
next  section,  but  rather  for  the  purpose  of  corroborating  the  force  equation 
used.  Furthermore,  friction  at  the  pivots  may  be  excluded,  since  the  pen- 
dulum is  usually  in  motion,  swinging  about  its  position  of  equilibrium,  so  that 
friction  would  have  operated  both  ways.  There  remains  the  possibility  of  an 
excess  of  pressure  in  the  film  of  air  within  a  metallic  fissure  as  compared  with 
the  surrounding  air.  To  obtain  some  quantitative  data,  since  F'R  =  6s.2&N, 
one  may  note  that  the  average  values  of  2AA/"  were  roughly  as  follows: 

Table  6  .........................................  2AW=o.O4O  cm. 

Tables  1,2  ......................................  2A#=o.oi3,  0.022,  0.038  cm. 

Table  4  .........................................  2AJV=o.O4O  cm. 


Hence 

Maximum, 


65.2X0.020  =  1.3  dynes. 


Minimum, 


o.4  dyne. 


66     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


As  the  area  of  the  disk  is  324  sq.  cm.,  the  corresponding  average  pressure 
observed  is  therefore  here 

p  =  4X io~3  dynes/cm.*= 4X  to"9  atm.  =  3  X  io~7  cm.  Hg. 

in  the  maximum  case,  or  about  one-third  of  this  in  the  minimum  case.  These 
differences  are  to  be  referred  to  the  distance  apart  of  the  plates  which  was  not 
always  measured  in  the  earlier  experiments. 

The  maximum  of  gravitational  attraction  of  the  air  within  the  crevice  by 
either  disk  is  72  TTS  where  7  is  the  gravitational  constant  and  ?  the  mass  of 
the  disk  per  square  centimeter  or  78.5  X  1.4,  so  that  271-9  =  78.8  dynes  per  gram 

TABLE  6. — Equilibrium  of  brass  disks  after  long  waiting.    Constants  as  in  table  I. 


Remarks. 

A* 

lO'AW 

Time. 

Remarks. 

A* 

io»XAtf 

Time. 

Free 

cm. 

2.IO 

cm. 
119 

min. 
SQ 

Free  .. 

cm. 

2  IS 

cm. 

—21 

min. 
6 

I 

105 

11 

27 

7 

I 
3« 

i 
3 
5 
7 
9 
ii 

13 
17 

V 

-  8 

+   2 

6 

10 

13 
14 
14 
15 

8 

10 
12 
14 

16 

18 

20 
22 

22 

19 

Free  
VI 

2.IO 

1  02 

7O 

24 
26 

From  2.  05... 
Free  

2.10 

8 

o 

22 
24. 

45 

42 

28 
30 

II 

6 

26 

»9 

3* 

Free  

2-15 

—  21 

28 

17 

8 

Ill 

-  8 
-  6 

—    2 
+   2 
10 
II 
II 

30 
32 

3 

39 

41 
43 

Free  
VII 

0.75 

44 
40 
47 
54 
59 
53 

39 
41 

$ 

48 
50 

1 

Free... 

O.7O 

S6 

C| 

Contact  .... 
Free  

2-15 
7O 

75 

Aft 

VIII 

49 

53 

IV 

59 
59 

4° 
50 
53 

45 
52 

55 
57 

48 

55 

Free 

o  65 

78 

64 
50 

57 
60 

IX 

56 
56 

61 
63 

Contact  .... 

0.70 

-150 

47 
47 

65 
67 

of  air  attracted.  If  the  air-film  is  i  mm.  thick,  the  mass  per  square  centi- 
meter is  thus  o.i  Xo.ooia  =  10-*  nearly,  and  hence  the  pressure  increment  can 
not  exceed  £=78.8X10-*.  The  smallest  observed  pressure  is  thus  1/7  times 
in  excess  of  the  largest  computed  values.  The  forces  in  the  fissure  must  thus 
of  a  different  kind  from  increased  gravitational  attraction  if  the  excess  of 
air-pressure  in  the  field  is  to  account  for  the  observed  phenomenon 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     67 


36.  Plates  electrically  charged.—  The  endeavor  is  next  to  be  made  to  over- 
come the  repulsion  of  plates  by  aid  of  opposite  electrical  charges  placed  upon 
them.    The  experiments  at  present  will  necessarily  be  somewhat  crude,  since 
TABLE  7.—  Effect  of  electrical  potential.    Constants  as  in  table  i.     12.5  volts  applied. 


Remarks. 

A* 

lO'Atf 

Time. 

Remarks. 

A* 

lO'Atf 

Time. 

Potential  off.  . 
+  12.  5  volts  on  . 

Potential  off  .  . 
II 

—  1  2.  5  volts  on  . 
Ill 

Potential  off  .  . 
IV 

cm. 
0-75 
•75 

•75 

•75 
•75 

cm. 
65 

ii 

24 
28 
31 

37 
43 
47 
50 
32 

21 
20 
20 
25 
29 
32 

etc. 

min. 

42 
44 
46 
48 
50 
52 

3 

59 
61 

3 

7 
9 
II 

13 
15 

+  I2.5voltson  . 
VI 

Potential  off  .  . 
VII 

cm. 
.80 

.80 

cm, 

*32 
20 
II 
15 

39 
46 
51 
57 
54 

min. 

22 

11 

28 
29 
31 
33 
35 
37 

—  I2.5voltson  . 

.80 

*33 
20 
8 
II 

38 
40 
42 

Contad 

;  

.80 

—  101 

Mean  results. 

Volts. 

A* 

lo'A./V 

Contact           , 
i^AW 

Contact  

0-75 
•75 
.80 

-  9 
+63 
*5i 
57 
50 
55 

15 
17 
19 

21 

Potential  off  .  . 
Potential  off  .  . 
V 

H-  H- 

M  K> 

61  ocn  0 

0-75 

& 

.80 

64 
19 
50 
II 

—9          0.036 
(Dist.  .004) 
—  101          .075 
(Dist.  .050) 

Swinging. 


•05 
•03 
01 
-41 
~G3 
-05 

-47 
< 

"ft 

Iff 
i 

9  < 
1 

<# 

>>& 

> 

6 

no 

\ 

4 

h 

M 

f 

1 

W 

\ 

/> 

\ 

A 

<2-5t> 

'                S 

J  1 

v* 

J-6U 

\ 

4 

i 
i 

%) 

-n 

c, 

•8< 

-«• 

to 
ft 

i 

8 

i 

1 

i 

/ 

3 

t 

+\ 

i 

/»/    0 

s 

\ 

-> 

s. 

J1""    20      40      60      80      100     #0      O07*  -05      -10      & 

FIG.  41. 

the  pier  is  liable  to  warp  during  the  time  in  which  the  observations  are  made. 
Nevertheless  the  corroboration  obtained  is  of  great  value. 


68     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

The  observations  were  made  in  time  series,  when  the  plates  were  close 
together.  For  plates  farther  apart  this  is  not  essential,  but  in  the  absence 
of  other  than  the  air-damping  the  suspended  plate  oscillates,  so  that  mean 
values  have  to  be  taken.  It  will  be  necessary  to  await  the  summer  in  order 
that  all  observations  may  be  made  in  a  room  free  from  artificial  heat. 

In  table  7  and  fig.  41 A  I  have  inscribed  the  first  observations  from  plates 
close  together.  In  table  8,  the  summary  of  all  the  observations  is  given.  In 
series  i  to  4,  table  7,  the  plates  are  probably  not  far  enough  apart,  though 
the  contact  position  is  still  beyond  the  equilibrium  position  by  about  0.014 
cm.  It  was  not  necessary  to  wait  for  the  uncharged  position  of  the  plates,  as 
this  remained  pretty  constant  long  before  and  after  the  experiments.  In 
series  5,  6,  7,  and  8,  the  distance  apart  is  increased  about  0.042  cm.,  and  there 
is  no  danger  of  actual  contact  (free  space  0.056  cm.  when  charged),  so  that 
actual  repulsion  is  in  question. 

TABLE  8. — Summary. 


d 

Atf 

d' 

Vots. 

V'comp. 

FJW,. 

Fltfomf. 

Correc- 
tion AN 

FM 

F'R 

* 

y 

/ 

(I) 

cm. 

(2) 
cm. 

(3) 
cm. 

volts 

(5) 

volts. 

(6) 
dynes. 

(7) 
dynes. 

(8) 

(9) 

(10) 

cm. 

(ii) 

cm. 

(12) 

dynes. 

0.056 

0.039 

0.075 

12-5 

7-43 

7-i 

2-54 

O.IIO 

2.8 

0.036 

+  .020 

4-7 

.044 

.014 

.051 

6.2 

3-56 

2.85 

•94 

.044 

3-o 

.0146 

+.029 

1.90 

.102 

.021 

.112 

12.5 

9-93 

2-15 

1-37 

•033 

1-57 

.003 

+.096 

.46 

•133 

.Ol6 

.141 

12.5 

i  1-3 

1.27 

1.04 

.019 

1.22 

.001 

+.132 

•19 

.014 

•045 

.036 

12.5 

19.0 

114.0 

2-93 

1.75 

39- 

•85 

-.84 

no. 

Table  8  shows  the  essential  data  of  these  and  subsequent  experiments  in 
which  the  distance  d,  in  centimeters,  between  the  plates  is  gradually  increased. 
AAf  shows  the  difference  of  displacement  (in  centimeters)  observed,  when  plates 
were  respectively  charged  and  uncharged.  AAf  thus  measures  the  displace- 
ment of  the  suspended  plate  in  half  centimeters.  From  &N  the  approximate 
electric  force  (ignoring  the  repulsion  of  plates)  may  be  computed  as  above, 
F'R  =  6$.2hN.  This  is  given  under  F'R  computed,  in  the  seventh  column. 

Plates  were  charged  by  aid  of  a  storage  battery  to  the  potential  shown  under 
V<*s  (volts),  in  column  4.  From  V  and  d,  the  attraction  of  plates  F'R  may  be 
computed,  since 


A  =  irRi  being  the  area  of  each  plate.    The  results  are  given  in  column  6 
and  the  ratio  F/F'  in  column  9. 

Furthermore,  the  potential  V  may  also  be  computed  from  FR,  since 

V.I 

R 

and  this  is  inserted  in  the  fifth  column. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      69 

From  the  computed  values  of  F'R,  since  FR  is  equal  to  6s.2XAAf,  the  true 
value  of  the  displacement  in  the  absence  of  repulsion,  AAT.may  be  computed  as 


65.2 

the  result  being  given  in  the  eighth  column  in  centimeters.     It  shows  the 
corresponding  displacement  of  the  suspended  plate  in  half  centimeters. 

The  two  values  of  A/V  and  AA/7  now  give  us  the  displacement  due  to  repul- 
sion in  centimeters, 


as  shown  in  column  10.    Again,  the  distance  apart  (in  centimeters)  of  the 
uncharged  plates  d'  is  given  in  the  third  column,  being 


and  found  from  observation  directly.  Finally,  the  residual  distance  apart  of 
the  plates,  y,  if  the  suspended  plate  had  taken  its  true  displacement  A/V'  (in 
the  absence  of  repulsion),  is  given  in  the  eleventh  column,  since 


In  every  case,  except  the  first,  in  which  y  is  negative,  the  plates  when  charged 
at  a  distance  d  apart  were  not  under  forces  sufficient  to  put  them  in  contact. 
One  must  observe,  however,  that  for  a  distance  apart  y  when  d<y,  the  forces 
would  not  increase  correspondingly.  Only  in  case  5  is  d=y,  nearly.  Thus, 
without  repulsion,  the  disks  should  have  been  thrown  in  contact  when  charged, 
in  all  cases.  In  the  actual  presence  of  repulsion  this  was  not  observed,  except 
perhaps  in  the  first. 

The  substance  of  these  investigations  is  contained  in  column  9,  where  the 
ratios  of  F'R  computed  electrically  and  the  value  of  FR  from  independent  data, 
i.e.,  from  the  given  displacement  A/V  of  the  horizontal  pendulum,  are  given. 
It  is  seen  that  the  ratio 


decreases  as  the  charged  plates  are  farther  apart  (d),  until  at  d> 0.13  cm.,  the 
ratio  is  nearly  i ;  i.e.,  the  repulsion  of  plates  nearly  vanishes  when  their  dis- 
tance apart  markedly  exceeds  i  mm.  Just  how  large  d  would  have  to  be  in 
order  that  F'R/FR=i,  I  did  not  endeavor  to  find,  since  the  suspended  plate 
vibrates  annoyingly  for  large  distances  apart.  In  other  words,  definite  ex- 
periments of  this  kind  must  be  left  for  the  summer  months.  The  constants 
of  the  pendulum  should  then  also  be  determined.  Moreover,  in  a  lighter  pen- 
dulum, the  sensitiveness  may  be  indefinitely  increased,  particularly  when  the 
pendulum  is  provided  with  a  float,  while  the  error  due  to  the  inclination  of  the 
pier  does  not  simultaneously  increase,  an  obvious  advantage.  It  seemed  wise, 
therefore,  to  stop  the  work  for  the  present  at  the  point  of  progress  reached. 


70     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

Table  8,  however,  admits  of  a  number  of  preliminary  estimates  of  the 
decrease  of  repulsion  (f)  with  d,  the  distance  apart  of  plates,  for  we  may  write, 
/=  65.2  X2X=  130*  dynes,  nearly. 

These  values  are  given  in  column  12  of  table  8  and  in  fig.  416,  with  the  ex- 
ception of  the  first,  which  is  liable  to  be  anomalous  from  actual  contact.  The 
second  observation  also  seems  to  be  in  error  for  some  reason  not  detected. 
The  others  make  a  compatible  series.  The  forces  found  in  the  above  work 
(paragraph  10)  lay  between  1.3  and  0.4  dynes,  for  distances  of  the  same  order 
of  value,  but  which  were  not  quite  the  same  in  the  two  positions  of  the  fixed 
disks.  If  we  take  the  results  in  table  6,  which  are  probably  the  best,  AAT= 
(0.049—0.014)72=0.018  cm.;  d=  (0.031+0.097)72  =0.064  cm.;  /=6s.2X 
0.018=1.17  dynes,  the  results  of  /  and  d,  as  shown  by  the  cross  in  fig.  4iB, 
fit  in  very  well  with  the  present  data  obtained  from  electric  attraction.  The 
repulsion  therefore  has  throughout  been  found  of  the  same  order  of  magnitude. 
The  pressure  corresponding  to  the  above  thrust  /  is  found  (as  above)  by 
dividing  by  the  area  A  of  the  disks,  whence 


We  may  then  compute  the  attraction  of  the  disks  per  gram  of  air  film,  at 
a  distance  h  from  the  disk,  similarly  to  the  ordinary  case  of  the  barometric 
formula, 


or  - 


an 


Thus,  if  one  can  detect  the  variation  of  /  with  h,  the  molecular  attraction  of 
the  disk  per  gram  of  air  should  be  discernible. 

37.  Conclusion.  —  By  the  application  of  displacement  interferometry  to  the 
deviations  of  the  horizontal  pendulum,  I  find  that  two  parallel  rigid  plates 
whose  distance  apart  is  of  the  order  of  i  mm.  and  less  repel  each  other,  in  air, 
with  a  force  far  in  excess  of  their  gravitational  attraction.  This  force  in- 
creases rapidly  (certainly  as  fast  as  the  inverse  square)  as  the  distance  of  the 
plates  decreases,  and  vice  versa,  but  can  be  recognized  beyond  a  millimeter  of 
distance.  For  brass  plates  20  cm.  in  diameter  and  i  mm.  apart,  the  repulsion 
in  question  is  of  the  order  of  0.5  dyne  and  therefore  equivalent  to  a  pressure 
of  0.0015  dyne-cm,  or  roughly  io~9  atmosphere.  It  is  in  excess  of  any  electric 
repulsion  due  to  the  absolute  voltaic  potential  of  the  disks.  The  suspended 
plate  reaches  its  position  of  equilibrium  gradually,  the  motion  progressing  at 
a  retarded  rate  through  infinite  time,  in  a  way  characteristic  of  the  viscosity 
of  the  film  of  air  between  the  plates. 

I  have  estimated  the  intensity  of  the  force  both  from  the  repulsions  of  a 
vertical  plate  suspended  from  the  horizontal  pendulum  on  opposite  sides  of  a 
fixed  parallel  identical  plate;  also  by  charging  pairs  of  plates  to  a  given  differ- 
ence of  potential  for  a  given  distance  apart.  So  far  as  can  be  seen,  the  repul- 
sion is  caused  by  the  condensation  of  air  on  the  surface  of  the  plates  by  molec- 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      71 

ular  and  not  by  gravitational  force  (which  is  too  small).  Hence,  the  method 
employed  should  enable  the  observer  to  find  the  density  of  the  concentration 
in  terms  of  the  distance  from  the  plate  and  the  law  of  attraction  of  the  plate 
in  terms  of  distance  within  the  small  distances  in  question.  In  other  words, 
a  method  for  direct  investigation  of  molecular  force  is  here  apparently  given. 

Correction:  The  effect  of  gravity  in  Chapter  II  was  overestimated  and  the 
data  have  been  withdrawn. 

To  make  an  estimate  of  the  gravitational  attraction  between  the  largest 
plates,  it  is  sufficient  here  to  consider  the  attraction  of  contiguous  disks 
under  the  approximate  form 

m*  w? 

F  = 


where  ^  =  6.7X10-*  is  Newton's  constant,  a  the  density,  ^  =  468  g.  the  mass, 
A  the  area,  r  =  10  cm.  the  radius  of  the  disks.    We  may  then  write 

,#£ 

m        M  _ 
~2r  r2  g<ph(i+mR/Mh) 

and  on  inserting  the  value  of  the  variables  <p  =  •  01  1,  h  =  80  cm.,  M  —  1,250  g., 
R  =  in  cm. 

AAT=4Xio-6, 

equivalent  to  a  little  over  one-tenth  of  a  vanishing  interference  ring.    Hence 
the  gravitational  attraction  could  not  have  been  recognized,  as  it  was  not. 


CHAPTER  III. 


THE  REFRACTION  OP  LONG  GLASS  COLUMNS  MEASURED  BY 
DISPLACEMENT  INTERFEROMETRY. 

38.  Introductory.  —  The  measurement  of  indices  of  refraction  and  their 
differences  for  different  colors,  in  terms  of  the  shift  of  the  ellipses  in  the  spec- 
trum, seemed  to  give  an  opportunity  for  unusual  sensitiveness  of  method 
when  long  columns  of  glass  are  inserted  in  one  of  the  interfering  beams.  But 
this  expectation  was  not  realized  in  full,  as  the  amount  of  shifting  per  unit  of 
displacement  of  the  micrometer  mirror  decreases  with  the  thickness  of  the 
glass,  or  the  length  of  the  column.  The  measurements  made  are  nevertheless 
interesting  as  a  test  of  the  availability  of  the  equation 


where  Nc  is  the  coordinate  of  the  movable  mirror,  corresponding  to  the  center 
of  the  ellipse  at  the  wave-length  X  of  the  spectrum.  R  is  the  angle  of  refrac- 
tion, ft  the  index  of  refraction  of  the  glass  for  the  same  color  (the  angle  of 
incidence  at  the  grating  being  7),  and  e  the  thickness  of  the  glass  column  in 
the  direction  of  the  penetrating  ray.  For  most  purposes  the  Cauchy  equation 
may  be  used  for  determining  dn/d\. 

39.  Glass  columns.—  The  column  to  be  tested  is  placed  with  its  faces  normal 
and  symmetrical  to  one  of  the  component  beams,  the  corresponding  mirror 
having  been  advanced  proportionately  to  the  length  of  the  column,  until  the 
ellipses  appear.  The  fine  adjustment  is  then  made  with  the  micrometer  screws, 
the  displacement  AAT  needed  to  shift  the  centers  of  the  ellipses  from  one  spec- 
trum line  to  the  next  in  succession  being  observed.  This  is  the  chief  datum 
of  interest  in  the  present  paper;  for  from  it  the  tests  in  question  may  be 
constructed. 

The  first  columns  provided  were  made  from  thick  glass  rods,  the  ends  of 
which  had  been  ground  off  normally  to  the  axis  of  the  cylinder  by  Mr.  Petit- 
didier.  But  in  none  of  the  rods  prepared  was  the  glass  sufficiently  homogene- 
ous to  admit  of  its  use.  In  some  cases,  in  fact,  the  stress  or  its  equivalent 
was  so  strong  as  to  make  the  rod  virtually  opaque  and  the  polarization  figures 
correspondingly  intense.  The  attempt  was  made  to  remove  the  stress  by 
annealing  at  low  red  heat,  but  this  also  was  quite  unsuccessful. 

Failing  in  all  other  trials,  I  finally  resolved  to  build  up  columns  from  ordi- 
nary plates  of  glass,  cemented  together  with  Canada  balsam.  In  this  way  I 
72 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      73 

obtained  adequately  clear  columns,  though  the  color  of  the  glass  is  frequently 
a  disagreeable  feature.   The  three  columns  made  had  the  following  dimensions : 


Length. 

Breadth. 

Height. 

No.  of 
plates. 

Average 
thickness 
of  each. 

Bluish  glass 

cm. 

25  4. 

cm. 
g 

cm. 

cm. 

Greenish  glass  

22.87 

6 

Greenish  glass  

7.l675 

6 

IO 

In  order  to  keep  the  plates  together,  the  top  and  bottom  of  the  long  column 
AB,  fig.  42,  were  provided  with  close-fitting  strips  of  metal  mm  and  m'm', 


JW 


FIG.  42. 

tightened  by  the  screws  ss'  on  both  sides.  It  was  then  incased  in  a  wood  and 
metal  sheath  CD,  carrying  a  long  stem  5  at  right  angles  to  the  length  of  the 
column,  by  aid  of  which  it  could  be  clamped  in  any  necessary  position  rela- 
tively to  the  interferometer. 

Over  half  a  year  was  allowed  for  the  aging  of  these  columns,  at  the  end  of 
which  time  the  balsam  was  hard  (the  column  having  been  kept  under  stress), 
quite  clear,  and  free  from  air-bubbles.  In  fact,  the  narrow  beam  AB  from  the 
collimator,  passed  through  these  columns  twice,  came  to  an  adequately  sharp 
focus  in  the  telescope,  and  no  difficulty  in  finding  or  adjusting  the  interference 
rings  was  experienced. 

In  view  of  the  large  glass  path  the  ellipses  were  necessarily  quite  small  and 
their  motion  in  response  to  the  micrometer  screw  sluggish.  They  were  con- 
tinually in  motion,  owing  to  the  unavoidable  tremors  to  which  the  laboratory 
is  subject,  indicating  of  course  that  the  evanescence  of  rings  is  still  commen- 
surable with  the  wave-length  of  light,  whereas  the  sensitiveness  of  the  shift 
has  been  reduced  in  proportion  to  the  length  of  column.  Only  in  one  respect 
was  the  behavior  peculiar.  It  was  quite  common  to  obtain  only  the  right- 
hand  half  or  the  left-hand  half  of  the  set  of  sharp  concentric-ring  patterns; 
i.e.,  on  one  side  of  the  vertical  line  passing  through  the  center  of  ellipses  the 


74     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

interference  pattern  was  quite  absent,  while  it  was  clear  on  the  other.  This 
seemed  to  be  particularly  the  case  whenever  the  component  ray  did  not  quite 
retrace  its  path,  i.e.,  when  the  paths  through  the  column  were  separated;  but 
I  have  been  unable  to  find  a  reason  for  the  peculiar  result. 

40.  Equations. — In  the  earlier  papers  I  showed  that  the  coordinate  of  a 
movable  opaque  mirror  Nc,  varying  with  the  position  of  the  center  of  the  ellip- 
ses in  the  spectrum,  in  case  of  the  displacement  micrometer,  could  be  written 


where  e  is  the  thickness  of  the  plate  of  the  grating,  or  other  plane  parallel 
glass  body  interposed  in  one  of  the  component  beams,  R  the  angle  of  refraction 
(the  corresponding  angle  of  incidence  is  I  for  all  colors)  for  the  color  X  and  the 
index  of  refraction  n.  If,  in  addition  to  the  plate  of  the  grating,  the  column 
of  glass  of  length  E  is  interposed  at  normal  incidence,  R  =  o,  but  having  the 
same  n  for  the  same  X, 

W  Ar. 


For  an  air-column  of  length  E  replacing  the  glass  column  for  the  same  color 

(3)  Ar. 
Hence, 

(4)  AN 


where  the  centers  of  ellipses  are  brought  to  the  same  spectrum  line,  both  for 
the  glass-  and  for  the  air-column.  Hence,  if  we  write  approximately,  for 
brevity, 

(5) 


In  other  words,  to  measure  the  index  of  refraction  /*  for  a  given  color  X,  the 
correction  -2EB/X*  must  be  added  to  the  corresponding  value  of  AAT0  the 
shift  of  micrometer,  or 

(7)  ^.W.-iEB/V 

Agam,  if  for  the  same  column  the  displacement  of  the  micrometer  Wt-Ne 
-N  c,  corresponding  to  different  lines  of  the  spectrum,  is  in  question,  since 

(8)  W. 


the  reduced  equation  becomes 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      75 

so  that  B  may  be  computed  from  this  equation;  or  if  three  terms  of  the  right- 
hand  member  be  taken  and  further  observation  made,  B  and  C  may  be  com- 
puted without  reference  to  &NC;  i.e.,  without  a  knowledge  of  the  displacement 
due  to  the  length  of  the  column,  for  any  color  X,  as  a  whole.  With  regard  to 
ANC,  I  may  recall  that  this  quantity  is  the  difference  of  the  air-paths,  from  the 
respective  points  of  intersection  of  the  normal  with  the  two  faces  of  the  grat- 
ing at  the  place  where  the  incident  white  ray  impinges,  to  the  two  opaque 
mirrors. 

41.  Equations.  Sensitiveness  in  terms  of  displacement—  To  the  same 
extent  in  which  equation  (i)  applies,  the  sensitiveness  dNJdK  may  now  be 
computed,  since 


and 

=  -Dcos6 


where  D  is  the  grating  constant  and  0  the  angle  of  diffraction  for  the  color  X. 
Performing  the  operations  and  reducing, 


dNe~cosR\ 
or,  by  inserting  equation  (5), 

(u)  ^L  =  JL 

dN.     zeB 

If  R  =  o  (normal  incidence), 

(15) 

or 

(16)  Ne- 

where  Nf  corresponds  to  X=  °°  ,  or  to  Cauchy's  constant  A.  From  equation 
(15)  it  appears  that  the  sensitiveness  (d\/dNc)0  is  inversely  as  the  thickness 
e  and  directly  as  the  cube  of  X,  for  a  given  glass.  Since  B/\*  depends  on  the 
refraction  of  the  glass,  d\/dNc  varies  as  \/e  and  as  i/(2?/X2).  It  is  this  feature 
that  makes  it  ultimately  unprofitable  to  use  long  columns.  No  additional  sen- 
sitiveness is  gained;  the  glass  absorbs  more  and  more  fully.  The  columns  are 
not  apt  to  be  homogeneous,  and  the  ellipses  become  excessively  small  and 
sluggish  in  their  motion.  They  offer,  however,  an  excellent  corroboration  of 
the  sufficiency  of  equation  (i).  Writing  this  in  the  form  (2),  equation  (17) 
may  be  deduced  on  adding  — 


<K_JW 

dN.     2B  \ 


so  that  the  predominating  term  for  long  columns  is  \*/6EB;  or  for  normal 
incidence,  accurately  \*/6(E+e)B. 


76      EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 
If  in  equation  (13)  we  convert  d\  into  dd 


whence 


dN~  DcosecosR 


we  may  compare  this  deviation  with  the  corresponding  case  when  X  and  n  do 
not  vary,  but  R  and  N  do;  i.e.,  the  case  of  motion  along  the  direction  of 
any  given  spectrum  line.  This  is  a  comparison  of  the  vertical  and  horizontal 
axes  of  the  ellipses. 

42.  Equations.    Sensitiveness  in  terms  of  order.  —  The  quantity  d\/dnt 
where  n  is  the  order  of  the  fringe  of  color  X,  has  subsidiary  interest. 
Since 


dn  ~  a  e  (cos  R-  X  sec  R  •  dp/dX)  -N 

generally,  and  the  condition  of  centers  of  ellipses  is  dl/dn  =  oo  ,  the  coordinate 
of  a  center  is 


If  we  combine  this  with  the  general  equation 

N—ett  cos  /?—  nJ/2 

using  «,  as  the  order  corresponding  to  centers  of  the  wave-length  X,  and  apart 
from  signs, 


cos  R  dl  =  cos  R# 

In  case  of  the  above  blue  column,  for  example, 
'=25.4  cm.;  B  =  4.6Xio-";  K=0; 

4X25.4X46X10-" 

"  --  =37' 


4°° 


half  wave-lengths  of  path  difference  in  the  glass,  not  including  the  addi- 
tional number  in  the  grating. 
It  is  interesting  to  return  to  the  original  equation 

dl=  __£  _  !_ 
dn       "a   Nc-N 

where  Nt-.  (cos  K-X  sec  Rfc/dx)  and  -dK=D  cos  6  d8,  since  sin  .'-sin  •- 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      77 

If  for  simplicity  we  put  7 = R  =  o  (the  actual  case  with  the  columns  in  ques- 
tion, incidence  being  normal),  cos  8  =  \/i—H'1/D*  and 


dn      aD  cos  6  N,-  N     2(Ne-N)VD*  -  A* 
If  the  center  of  ellipses  is  at  the  E  line 


To  find  the  size  of  the  fringes  at  any  other  line,  the  D  line,  for  instance,  we 
may  again  take  the  example  of  a  blue  column  (table  10  below)  where 
Af«—  AT=o.255  and  put  X  =  5.3Xicr*,  the  grating  space  D  =  2.5X10-*,  whence 


(d6\ 
— )    = 


28.1  X  IP"10 


2X.2SsXio-V6.2S-.28 


=  1 0-^X2 2. 6  radians  or  4. 6* 


At  the  D  line,  therefore,  the  distance  between  consecutive  fringes  would  be 
less  than  5  seconds  of  arc,  showing  the  diminutive  fringes  to  be  expected. 
After  leaving  the  center  the  fringes  become  more  and  more  nearly  equidistant. 
We  may  therefore  estimate,  if  the  angle  between  the  D  and  E  line  is  here 
0  =  4,380",  that  somewhat  less  than  4,360/4.6  =  950  fringes  would  be  encoun- 
tered between  D  and  E.  They  would  therefore  not  be  useful,  except  near  the 
center,  where  d\/dn  =  «> . 

43.  Observations.  Green  glass  column. — In  spite  of  the  clearness  of  the 
column,  the  light  absorbed  at  the  ends  of  the  spectrum  makes  it  nearly  im- 
possible to  recognize  the  small,  sluggishly  moving  ellipses.  The  observations, 
therefore,  are  reasonably  good  only  between  the  D  and  E  lines.  In  some  cases, 
moreover,  it  is  easy  to  mistake  the  lines,  from  the  coincidence  of  the  direct 

TABLE  9.— Green  column.   £  =  22.87  cm.;  «  =  o.68  cm.;  B  =4.6X10-";  1  =  15°;  £=9.7°: 
=sin  7/sin-R;  (3-E+«  cos  R +2e/cos  R)  =70.66;  M  =  i-53- 


Lines. 

C-D. 

D-E. 

E-b. 

6-? 

b-F. 

6N,  

0.185 

0.235 

0.046 

O.I  12 

.166 

•233 

50 

.122 

.... 

.168 
•175 

.236 
•237 

48 
40 

.065 
67 

.... 

.181 

.232 

•045 

•075 



.181 
.175 
•175 

.... 

48 
43 
45 

.084 

70 

67 

'.'.'.'. 

.174 



43 





I7d 

Mean  SNf  observed  

•  1754 

.2346 

.0453 

SNf  computed  

.i8n 

.2347 

.0442 

0.1610 

78     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

spectra  of  higher  orders,  with  the  two  interfering  spectra.  For  work  of  this 
kind  it  would  have  been  preferable  to  make  I  (angle  of  incidence)  about  45°. 
The  individual  observations  are  given  in  table  9.  Between  D  and  E  the  error 
is  about  ±0.002  on  a  length  of  i2.6=AA/"e  for  the  total  displacement;  i.e., 
about  1.5X10-*  of  M—  *  •  IQ  computing  8NC  from  equation  (9),  the  value  of 
B  computed  for  light  crown  glass  from  Kohlrausch's  tables  was  accepted, 
as  a  special  measurement  for  so  many  plates  of  glass  seemed  out  of  the  ques- 
tion. The  films  of  Canada  balsam  are  negligible. 

The  quantity  AATC  was  roughly  measured.  To  find  this  accurately  it 
would  have  been  necessary  to  make  special  adjustments,  as  it  is  large  (AA/"C  = 
1  2.  6  cm.,  quite  above  the  range  of  the  micrometer),  and  as  a  readjustment  of 
the  mirror  must  be  made  in  the  presence  and  absence  of  the  column,  for  which 
it  is  difficult  to  make  an  allowance.  The  end  faces  were  not  quite  plane 
parallel.  Using  equation  (6)  the  value  of  the  correction  zEB/\*  is  0.605  cm., 

whence 

2£/#  =  1.525 


at  the  D  line.  The  value  found  directly  from  the  total  reflection  for  a  similar 
glass  was  1.521.  As  without  the  correction  ^=1.551,  the  corroboration  of 
the  equation  is  adequate.  One  may  note  that  —  zB/\*  =  —  0.0265,  a$  a  correc- 
tion of  n,  is  independent  of  E,  the  length  of  the  column.  But,  for  purposes 
like  the  present,  a  small  thickness  of  glass  (E  about  i  cm.  and  within  the  range 
of  the  micrometer  screw)  is  preferable,  even  if  the  accuracy  could  be  enhanced 
by  using  a  stronger  telescope. 

Table  9  shows  that  the  computed  values  of  SNC  happen  to  coincide  with 
the  observed  values  between  the  D  and  E  lines.  Between  D  and  C,  D  and  6, 
the  results  are  quite  within  the  errors  of  observation  and  satisfactory.  The 
F  line  was  obviously  not  observed,  some  other  line  in  this  dark  part  of  the 
spectrum  being  mistaken  for  it.  Thus  the  line  X  =  49.  5  8  would  give  dNc  =  o.ioS, 
the  line  X  =  50.41  would  give  0.064,  each  coming  close  to  some  of  the  observa- 
tions. The  results  as  a  whole  therefore  attest  the  accuracy  of  the  equation 
used,  as  the  computed  lines  are  clearly  better  than  the  observed  lines. 

44.  Observations.  Blue  glass  column.  —  Although  this  column  was  more 
colored  than  the  other,  the  observations  were  apparently  not  inferior.  Table 
10  contains  the  results.  It  was  just  possible  to  reach  the  F  line,  visually.  As 
before,  these  data  reduce  largely  to  the  shift  from  D  to  E.  The  column  was 
too  long  to  be  compassed  by  the  contact  lever  used,  and  the  length  E  given  is 
therefore  approximate. 

To  compute  M  from  AAre=i4.o  cm.,  observed,  the  equation  is  as  before 


where  the  last  term  is  0.0265  for  the  same  B  and  X;  hence 

/i  =  1+0.5503  -0.0265  =  1.5248 
agreeing  sufficiently  with  the  experimental  result. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      79 


The  observed  and  computed  values  of  8NC  show  the  same  agreement  within 
the  errors  of  observation  as  before.  Differences  are  due  to  the  value  of  B 
used,  which  is  naturally  not  quite  the  same  as  in  the  preceding  case  and  should 
here  have  been  about  2  per  cent,  smaller.  Between  b  and  F,  io6X  =  50.41 
would  give  8NC  =  0.0709,  which  was  probably  observed. 

TABLE  10. — Blue  column.    £=25.4  cm.;  «=o.68cm.;    5=4.6X10-";  7  =  15°;  -£=9.7°: 


Lines. 

C-D. 

D-E. 

E-b. 

6-?. 

b-F. 

SNc 

0.178 

0.2=^2 

O.O5O 

0.063 

O  157 

.218 
.196 

•254 
•255 
•257 
.256 
•254 
.256 

53 
47 
49 
47 
50 
48 

8 

.191 

Mean  6Ne  observed  

.197 

.2549 

.0491 

.174 

SNf  comouted  .  .  . 

.200S 

.2<%QQ 

.0400 

.1782 

45  Observations.  Shorter  column. — The  results  for  this  column  are  given 
in  table  n.  It  was  less  than  one-third  as  long  as  the  other  columns,  but, 
absorbing  less  light,  all  the  lines  were  seen.  The  ellipses  being  more  mobile, 
sharper  adjustment  is  implied;  but  the  F  line  could  not  be  recognized  with 
certainty  and  there  was  difficulty  at  the  C  line. 

In  this  case,  ^Nc  =  ^.g6i  cm.  lay  within  the  compass  of  the  micrometer. 
The  only  error  therefore  is  the  intermediate  readjustment  of  mirror  in  presence 
and  in -absence  of  the  column.  The  index  of  refraction  of  the  D  line  is  thus 


M=I+  3.961 

7-1675 

which  is  of  the  same  order  as  before. 


1.5261 


TABLE  n.  —  Short  column.    £  =  7.1675  cm.;  «  =  o.68  cm.;  B=4.6Xio- 


15°;  -8=9.7°; 


Lines. 

C-D. 

D-E. 

E-b. 

fr-?. 

b-F. 

&NC  

0.0588 

0.0795 

0.0147 

0.0432 

0.0560 

589 

790 

158 

353 

558 

591 
576 

792 

785 
784 

153 
155 
ISO 

•0305 
319 
323 

560 
571 
548 

.0224 

230 

213 

331 

Mean  SNe  observed 

.0586 

.0789 

.0153 

.0559 

SNf  computed  

.06035 

.07821 

.01473 

.... 

•05363 

The  results  for  this  series  are  scarcely  as  good  as  the  preceding,  relatively, 
since  finer  micrometric  measurement  was  required;  but,  absolutely  considered, 


80     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

they  are  better.  Thus  between  D  and  E  the  agreement  is  within  7  X  io— 4  cm. 
It  is  obvious  that  between  b  and  F  different  lines  were  sighted,  some  of  them 
possibly  due  to  superimposed  direct  spectra.  Thus  for 

io«X= 50.41;  =  49.58  SNe= 0.02 13 1  =  0.0359;  etc. 

Both  these  and  other  lines  seem  to  have  been  used. 

46.  Summary. — The  results  contained  in  tables  9,  io,  u,  reproduced  in 
fig.  43,  show  that  equation  (i)  above,  or  any  of  its  derivatives  (4)  and  (8), 
gives  an  accurate  account  of  the  motion  of  the  center  of  ellipses  throughout 
the  spectrum,  even  in  case  of  such  extreme  conditions  as  are  introduced  by 
glass  columns  io  inches  or  more  long.  The  constants  of  Cauchy's  or  any  simi- 
lar dispersion  equation  may  therefore  be  obtained  directly  from  observations 
of  this  character.  In  such  a  case  a  linear  interferometer,  i.e.,  one  in  which 
I—R=Q  approximately,  would  be  specially  convenient.  If  8  refers  to  the 
difference  of  the  variables  for  2  lines  X  and  X' 


which  in  case  of  the  simple  dispersion  equation  gives  $B(E+  e)&-jp-    As  this 

linear  interferometer  will  have  other  interesting  properties,  it  has  been  thought 
worth  while  to  construct  it  in  connection  with  the  present  work. 


•4 

•s> 

•2 
-4 

0 

•4 

-2 

^ 

^ 

4 

I. 

T^ 

K 

-4 

^ 

X 

^^ 

---. 

^\ 

S3 

,t 

^^; 

^ 

^fe 
^ 

1  —  -« 

f 

TdxKf- 

•*• 

-< 

*^! 

^ 

9 
X^ 

4850&H54-56&8606264- 
FIG.  43- 

The  expectation  of  reaching  great  sensitiveness  by  using  long  columns  was 
not  fulfilled  in  view  of  equation  (17),  which  shows  that  the  ellipses  become 
more  and  more  sluggish  in  their  motion  through  the  spectrum,  as  the  column 


CHAPTER  IV. 


PART  I—  EXPERIMENTS  BEARING  ON  THE  PROPERTIES  OF  CORONAS. 

47.  Introductory.  —  There  are  a  number  of  obscure  points  in  the  theory  of 
coronas  when  the  particles  producing  them  range  in  size  from  about  io~3  cm. 
to  10  ~4  cm.  in  diameter.  These  relate  chiefly  to  the  colored  central  disks  and 
to  the  color  which  for  very  fine  particles  spreads  uniformly  over  the  white  source 
of  light.  In  the  latter  case  the  colors  are  strictly  axial  and  they  suggest  the  in- 
terferences due  to  thin  plates.  At  least  a  tentative  explanation  along  these 
lines  seems  available.*  Light,  moreover,  is  abundantly  reflected  by  the  par- 
ticles, as  may  be  tested  by  using  a  Nicols  prism.  It  seems  reasonable,  there- 
fore, to  assume  that  in  spite  of  their  small  size  the  light  is  also  transmitted  and 
that  the  effect  is  appreciable  when  the  column  of  fog  is  long  enough  in  the 
direction  of  the  impinging  light.  All  of  this  is  in  accordance  with  the  condi- 
tions under  which  axial  colors  are  produced.  If  they  were  regarded  as  dif- 
fractions within  the  geometric  shadows  of  the  droplets  whose  diameter  d  is 
decidedly  smaller  than  io~3  cm.,  the  axial  distance  6  in  front  of  the  droplet 
corresponding  to  the  color  X  would  be  b=dz/n\  nearly,  for  the  fringe  of  the 
nth  order.  Hence,  even  in  case  of  n=  i,  b  would  be  much  less  than  0.2  mm., 
whereas  the  axial  colors  are  seen  for  all  values  of  b;  i.e.,  they  do  not  vary 
with  6,  however  large  it  may  be  taken. 

The  disk  colors,  however,  belong  to  the  phenomenon  itself.  If  the  element- 
ary equation  for  a  single  particle  were  true,  i.e.,  if 


where  9  is  the  angle  of  diffraction  for  the  wave-length  X  and  the  diameter  of 
particle  d,  C  the  constant  given  by  Airy's  series,  and  s/R  the  aperture  of  the 
corona  shown  by  the  goniometer,  the  disks  should  invariably  be  white  and  red 
edged,  as  is  the  case  of  relatively  large  particles  and  small  coronas.  Actually, 
however,  the  white  disk  is  more  and  more  evanescent  as  d  is  smaller,  the  color 
being  particularly  vivid  in  case  of  the  green  coronas,  where  the  disk  is  almost 
quite  green.  The  disk  and  annuli  thus  recall  the  appearance  of  the  rotary 
polarization  of  a  quartz  crystal  cut  normal  to  the  axis,  though  of  course  all 
polarization  is  strictly  absent  in  the  colored  diffraction  phenomenon.  I  have 
in  fact  endeavored  to  identify  the  colors  by  the  aid  of  a  rotary  polariscope, 
fig.  44,  B  and  A  being  the  polarizer  and  analyzer,  Q'  the  quartz  rouge,  FC 
the  fog-chamber  at  a  distance  from  Q',  Q"  a  quartz  column  sufficiently  long 
to  give  a  white  field.  Hence  the  coronas  could  be  seen  directly  through  Q", 

*  Barus,  Am.  Journal  of  Sci.,  xxv,  1908,  pp.  224-226. 

81 


82     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

whereas  the  color  from  Q'  appeared  contiguously  on  the  left,  the  apparatus 
being  used  like  a  half-shade.  But  the  attempt  was  not  of  practical  value  for 
incidental  reasons. 

.  6  , 

\          <J$  ~^  w\  %-y f      c) 


FIG.  44- 


FIG.  45- 


48.  Experiments  with  a  grating. — Since  in  the  observation  of  coronas  the 
diffractions  toward  the  right  of  a  group  of  particles  on  the  left  are  coordi- 
nated with  the  diffractions  toward  the  left  of  a  group  of  particles  on  the  right, 
it  seemed  interesting  to  endeavor  to  reproduce  the  coronal  phenomenon  by 
two  identical  and  coplanar  gratings  with  their  rulings  in  parallel  and  at  a 
sufficient  distance  apart.  The  very  large  dispersion  of  the  usual  commercial 
grating  would  here  be  an  annoyance;  but  the  circular  grating  constructed  by 
Mr.  Ives,  with  5,000  lines  to  the  inch,  is  in  every  way  peculiarly  adapted  for 
comparison.  If  such  a  grating  is  placed  in  a  cone  of  somewhat  divergent 
white  light  from  a  lens,  an  interesting  succession  of  colors  appears  when  the 
axis  of  the  cone  intersects  a  white  screen  one  or  more  meters  off.  When  the 
grating  is  near  the  vertex,  the  axis  is  white;  as  the  distance  between  grating 
and  vertex  gradually  increases,  the  corona  shrinks  and  eventually  a  colored 
central  disk  appears.  The  reds  are  vague;  but  the  green,  blue,  and  violet 
disks,  seen  in  succession,  are  very  strong,  each  corresponding  to  a  particular 
distance  of  the  screen.  The  close  agreement  of  this  occurrence  with  the  disk 
colors  of  coronas  is  striking  and  seemed  worth  further  investigation.  It  is 
necessary  that  the  whole  grating  be  illuminated,  as  the  edges  are  largely 
responsible  for  the  phenomenon,  while  the  successive  annular  regions  modify 
it.  Hence,  an  annular  method  of  illumination  suggests  itself,  the  annulus 
being  concentric  with  the  circular  rulings.  In  case  of  sunlight,  a  breadth  of 
annulus  of  about  i  mm.  for  the  diameter  of  several  centimeters  gave  good 
results. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      83 

The  adjustment  adopted  is  shown  in  diagram  in  fig.  45^,  where  L  is  the  lens 
about  6  inches  in  diameter,  with  its  focus  at  F,  about  half  a  meter  or  more 
from  the  lens.  G  is  the  circular  grating,  5  the  distant  screen,  several  meters 
off.  On  the  plane  G  the  diameter  of  the  concentric  annulus  is  zy;  on  the  plane 
5  the  diameter  is  2*,  where  R\  is  the  distance  apart  of  the  planes  measured 
along  the  white  rays,  which  are  elements  of  a  hollow  cone  with  its  apex  at 
F.  If  E  is  the  center  of  the  ring  2X  and  R  the  normal  to  G,  the  rays  R'2  are 
diffracted  to  E,  and  with  R'i  determine  the  angles  0  and  i.  We  may  then 
write  in  succession,  for  first  order  of  spectra,  if  the  grating  space  is  D, 

(1)  sin*+sin0=D/A 

(2)  sin  i=(x—  y)  /R\    sin  9  =  y/R'* 

If  the  dispersion  y  is  small  compared  with  the  distance  R  and  the  incidence 
*  of  small  obliquity, 

(3)  R'i  =  R* 
whence 

(4)  x/K-l/D 

Thus  the  result  is  independent  of  the  diameter  of  the  annulus  2y,  if  R"  is 
small,  and  the  grating  may  therefore  be  placed  at  the  focus  F. 
The  equation  may  be  more  correctly  written 


y  being  always  small  compared  with  x.    If  R  is  the  normal  distance  between 
screen  and  grating,  the  full  expression  would  be  inconveniently 


or  if  y  =  o,  with  G  at  F, 


In  table  12  a  series  of  such  measurements  is  recorded,  in  which  X  is  computed 
from  D,  x,  y,  R.  Only  the  outer  lines  of  the  circular  grating  G,  fig.  450,  were 
used,  the  center  being  rendered  opaque  by  a  concentric  disk  of  cardboard  S'. 
The  grating  was  then  moved  into  the  divergent  cone  of  white  light  limited 
by  the  circular  hole  in  the  screen  S",  until  the  desired  color  appeared  on  the 
distant  white  screen  at  the  center  of  the  white  ring.  The  diameter  2x  of  the 
latter  is  then  the  only  variable.  Later  an  annulus  cut  in  an  opaque  screen 
5",  fig.  45&,  was  also  used.  In  such  a  case,  the  position  5"  determines  x,  while 
the  position  of  the  grating  is  of  no  consequence.  The  wave-lengths  so  obtained 
are  as  a  rule  larger  than  the  normal  values  expected,  a  result  due  in  part  to 
errors  in  the  judgment  of  color  and  in  part  to  the  approximate  equation  used. 
It  is  not  worth  while  to  enter  further  into  the  reason  for  this,  the  chief  point 
being  that  with  the  use  of  an  annular  source  all  the  colors  may  be  made  to 


84     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

appear  strongly  in  the  center  of  the  figure;  i.e.,  in  the  center  of  a  white  non- 
diffracted  ring.  If  y  is  practically  zero,  these  colors  will  at  first  sight  seem  to 
be  axial  colors  and  would  appear  to  be  equivalent  to  those  of  the  coronas  of 
the  fog-chamber.  This  does  not,  however,  seem  to  be  the  case,  for  reasons 
presently  to  be  given. 

TABLE  1 2.— Disk  colors  of  an  annular  grating.     D  =  5  X I  <r*  cm. ; 
X  =0.714*  (iy/x).+ 


Disk,  2y 

Color. 

White  ring 
*. 

Computed 
XXio«. 

6.5  cm  ... 

A    C 

Red  .  . 
Orange   .  . 
Yellow    .. 
Green.    .. 
Blue..    .. 
Violet.    .. 
Red  

cm. 
95 
86 
81 
73 
67 
57 
95 

70 
64 
60 
54 
50 
43 
69 

Orange  .  .  . 
Yellow  .  .  . 
Green  .... 
Blue  
Violet.... 

87 

83 

% 

59 

64 
61 
54 
49 

44 

Parallel  ra 

ys.    x~y. 

zy. 

Color. 

R. 

XXio«. 

A    C 

Violet 

*\O 

38 

&:::::: 

Violet.... 

39 

42 

It  follows  from  equations  (5)  and  (6)  that,  since  D  and  R  are  given,  X  varies 
as  x,  the  radius  of  the  white  ring;  therefore  also  with  y,  the  corresponding 
radius  of  the  white  ring  at  the  grating,  if  the  position  of  the  grating  is  fixed. 
For  x/R=y/R",  nearly.  Hence,  if  the  outer  rings  contribute  a  red  center, 
the  inner  rings  would  contribute  a  blue  center,  the  superposition  of  all  colors 
being  white  light;  i.e.,  the  usual  white  disk  of  these  grating  coronas.  On  the 
other  hand,  if  the  outer  rings  contribute  a  corona,  the  inner  rings  can  only 
contribute  blues  and  violets,  for  there  are  no  other  first-order  colors.  Hence, 
greenish-blue  and  violet  centers  may  be  produced  from  the  grating  as  a  whole, 
without  an  annular  source  of  light  (as  is  in  marked  degree  the  case),  whereas 
the  red  axial  colors  occur  only  when  the  source  is  annular. 

It  is  interesting  to  note  that  when  the  central  color  is  violet  the  reds  have 
already  overstepped  the  center  and  are  approaching  the  white  ring.  Fig.  46 
shows  that  when  the  divergence  of  the  undiffracted  white  annulus  is  not  too 
great,  it  is  easy  to  produce  the  internal  and  the  external  annular  spectrum  on 
the  outside  of  the  white  annulus.  In  such  a  case,  x'  denoting  the  distance  of 
the  red  annulus  from  the  center  of  the  screen  5,  at  a  distance  R  from  the 
grating  G, 

x'+ 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      85 


which  for  x'=y  =  o,  reduces  to  the  above  values. 
colored  ring  coincides  with  the  white  ring, 


If  x=x'  and  y  —  o,  i.e.,  if  a 


When  the  red  center  is  in  adjustment,  the  distribution  of  colors  along  the 
axis  from  screen  to  grating  is  naturally  in  the  order  of  violet,  blue,  green,  etc., 
and  they  are  relatively  dull  second-order  effects. 


FIG.  46. 


FIG.  47. 

49.  Experiments  with  small  coronas. — To  reproduce  these  phenomena  ob- 
jectively with  coronas  is  not  difficult,  so  long  as  the  coronas  are  of  small 
aperture,  like  those  obtained  from  lycopodium  spores.  If  the  adjustment  in 
fig.  456  is  used,  the  annulus  should  be  sharp  on  the  screen.  The  dusted  plate 
G  maybe  placed  anywhere  near  the  focus  F,  showing,  moreover,  that  the  thick- 
ness of  the  fog-chamber  would  not  be  effective.  The  best  results  were  obtained 
by  using  a  pair  of  lenses  of  long  focal  distance,  together,  as  in  fig.  47,  where 
A  is  the  annulus,  L  and  L'  lenses  of  focal  distances  225  and  120  cm.,  respec- 
tively, G  the  lycopodium  plate,  5  the  white  screen  at  a  distance  of  about  2 
meters.  By  moving  L  and  L'  together  or  the  reverse,  i.e.,  by  varying  their 
distance  D  apart,  the  diameter  ww'  of  the  white  ring  may  be  varied,  while  it 
is  kept  in  sharp  focus.  For  the  focal  power  is  i/F=C—C'D,  C  and  C'  being 
constant.  The  white  annulus  should  be  about  i  mm.  broad  and  5  cm.  in 
diameter.  Sunlight  is  preferable.  If  the  ring  is  too  thin,  the  colors  are  vague 
from  an  insufficiency  of  light. 


86     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

If  the  elementary  equation  of  §47  is  used,  the  results  will  not  differ  much 
from  those  of  the  grating,  provided  the  same  approximations  are  made.  For 
if  C  is  Airy's  constant,  we  have  in  succession,  d  being  the  diameter  of  the 
particle, 


(2)  sin  *  =  (x  -  y)  /R',    sin  0 

(3)  A 

If  y/R  and  (x-y)/R'  are  small  in  comparison  with  i  and  R'i  =  R'3,  practically, 

dx  dx 

(4) 


or  if  y=*o  and  the  plate  G  is  at  F, 

l-dx/CR' 

Using  the  equation  with  lycopodium,  a  difficulty  experienced  is  because  the 
colors,  on  projection,  are  reddish-brown  or  vague.  But  on  reducing  the  red- 
dish center  at  r  to  a  dot,  and  treating  it  as  a  blue  minimum,  X  =  49X10-*, 
the  following  values  for  the  diameter  of  the  spores  were  obtained : 

C  R'  x  d 

1.22        250  cm.        5.7  cm.        0.0026  cm. 
5-5  -0027 

This  seemed  to  agree  with  the  size  of  the  particles  seen  under  the  microscope. 
The  diameter  usually  quoted  is  0.0032  cm.  The  question  would  resolve  itself 
into  a  decision  of  the  color,  which  is  here  a  minimum  and  is  therefore  of  no 
further  interest.  If  we  regard  the  red  edge  of  the  disk  as  a  blue  minimum  and 
use  the  goniometer  of  chord  s  and  radius  R  =  $o  cm.,  s/2R=i.22\/d,  whence 
sd=73.2\  =  o.oo36.  In  my  experimental  work  with  fog  particles  I  used  a 
somewhat  smaller  constant,  ^5  =  0.0032. 

50.  Experiments  with  large  coronas,  annular  source.  Coronas  by  reflec- 
tion.— The  coronas  due  to  water  particles  are  not  intense  enough  to  be  avail- 
able for  projection,  unless  perhaps  a  fog-bank  more  than  12  inches  thick  is 
used.  The  screen  has  to  be  placed  so  close  to  the  fog-chamber  (R  very  small) 
and  the  color  varies  so  rapidly  along  R,  that  the  semblance  of  axial  color  (angle 
of  diffraction  zero)  found  above,  quite  vanishes.  The  particles  as  a  rule  range 
in  size  from  somewhat  above  io~J  cm.  to  somewhat  above  io~4  cm.,  after 
which  the  coronas  become  filmy.  The  work  must  therefore  be  done  subjec- 
tively, as  usual,  and  the  direct  light  to  the  eye  may  be  blotted  out  by  a  small 
screen  or  disk,  placed  between  the  point  source  and  the  eye.  In  such  a  case 
the  colors  beyond  the  edge  of  the  disk  are  very  vivid. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      87 

When  an  annular  source  is  used,  the  path  of  the  cone  or  cylinder  of  light 
through  the  fog  is  highly  illuminated  and  the  diffraction  is  both  inward  and 
outward  from  this  shell.  The  shadow  of  the  disk  of  the  annulus,  therefore, 
necessarily  remains  permanently  dark,  so  that  this  should  be  small  and  the 
cone  as  divergent  as  the  fog-chamber  permits.  The  colors  are  quite  brilliant, 
in  spite  of  the  fact  that  fewer  particles  are  used  than  in  the  case  of  an  unob- 
structed point  source. 

In  fig.  48,  if  FF  is  a  fog-chamber,  ww  and  w'w'  elements  of  the  conical  shell 
of  light,  a  position  of  the  eye  of  E  behind  the  fog-chamber  may  be  found,  at 
which  the  whole  interior  of  the  cone  flashes  into  the  uniform  color  of  the 
disk.  For  this  purpose  *  must  be  small  or  zero,  or  the  shell  nearly  cylindric, 
in  which  case,  if  R  is  the  distance  of  the  eye  from  the  center  of  the  chamber, 
s/R  the  aperture  of  a  given  color  minimum,  the  elementary  equation 
becomes 


In  this  case  s  is  constant  and  R  is  variable.    It  is  found  difficult  to  use  this 
method  practically. 

An  interesting  accompaniment  of  these  experiments  is  the  occurrence  of 
vivid  color  when  the  eye  is  in  a  proper  position  on  the  illuminated  side  of  the 
fog-chamber,  as,  for  instance,  at  E'  or  E".  In  other  words,  there  is  also  vivid 
diffraction  in  connection  with  the  reflected  light,  a  phenomenon  which  it  is 
difficult  to  detect  in  case  of  the  absence  of  the  annular  screen  A ,  since  all  the 
fog  is  illuminated.  These  colors  come  from  both  the  outside  and  the  inside  of 
the  cone.  When  the  colors  fade,  or  when  there  is  much  directly  reflected 
light,  they  may  be  restored  by  a  Nicols  prism,  as  this  on  orientation  cuts  off 
the  reflected  light  only.  The  diffracted  colors  are  in  no  case  polarized,  whether 
seen  by  transmitted  or  reflected  light. 


FIG.  48. 


Experiments  made  at  some  length  with  these  annular  sources,  supplied 
with  both  polarized  and  unpolarized  light,  did  not  lead  to  further  results 
worthy  of  note.  They  showed  clearly  that  the  axial  and  disk  colors  can  not 
be  explained  as  suggested  and  produced  above,  in  case  of  the  grating,  however 
many  points  of  resemblance  there  are.  In  fact,  if  with  regard  to  fig.  48  we 
write 


88     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


which  gives  a  rough  reproduction  of  the  facts,  the  aperture  of  a  corona  of 
average  wave-length  when  the  particles  approximate  IQ-*  cm.  must  soon 
exceed  90  °,  so  that  the  colors  are  spread  out  along  the  axis  of  a  shallow  cone 
close  to  or  within  the  fog-chamber.  It  would  require  a  similarly  divergent  cone 
of  light  to  make  these  colors  seem  axial.  Now,  in  producing  coronas  from  a 
single  small  source  of  light,  the  rays  which  strike  the  chamber  are  quite 
inadequately  divergent. 

Finally,  by  putting  the  electric  lamp  with  an  annular  screen  close  to  the 
fog-chamber,  thus  avoiding  all  lenses,  the  phenomena  were  produced  much 
more  brilliantly,  but  without  new  results. 


S2 


80 


# 


10^L 


g; 


-A 


I 


\ 


ncuL&t.^^ 


L 


4-      6       8      10     H 
PIG.  49. 


\ 


16      18 


51.  Coronas  from  a  point  source.— The  best  evidence  obtained  bearing  on 
the  nature  of  the  disk  colors  is  probably  found  in  my  own  earlier  experiments 
with  strictly  homogeneous  light.  The  green  light  of  a  mercury  lamp  is  used 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      89 

for  illumination,  as  the  source  must  be  very  intense.  The  results  obtained  are 
surprising  and  are  summarized  in  fig.  49  in  which  the  ordinates  5  are  the 
chords  of  the  radius  of  ^  =  30  cm.  subtending  the  aperture  28  of  the  coronal 
disk  or  ring  specified,  so  that  V2^  =  sin0.  The  abscissas  are  the  ordinal 
numbers,  z,  of  the  successive  partial  exhaustions,  all  of  them  identical.  Ar- 
ranged in  this  way,  all  the  apertures  s,  curiously  enough,  vary  linearly  with 
the  number  of  the  exhaustions  2,  while  the  fog-particles  are  gradually  increas- 
ing in  diameter  d,  exponentially.  If  the  quantity  5  were  laid  off  in  terms  of  dt 
the  curves  would  be  roughly  hyperbolic,  and  less  serviceable  for  exposition. 
The  three  curves  selected  from  many  similar  results  refer,  respectively,  to  the 
aperture  of  the  edge  of  the  green  disk,  and  to  that  of  the  inner  and  outer  edge 
of  the  first  green  ring,  the  latter  value  of  5  being  about  twice  as  large  as  the 
s  for  the  edge  of  the  disk.  During  the  earlier  exhaustions,  z=i,  2,3,  etc.,  the 
coronas  are  very  large  and  filmy,  and  a  sharp  value  of  5  is  out  of  the  question. 
Consequently  observations  in  the  regions  A  can  not  be  expected  to  agree 
closely.  The  graph  for  the  outer  ring,  moreover,  had  to  be  taken  from  a 
separate  series  of  observations. 

The  feature  of  these  experiments  is  this,  that  the  disk  and  the  first  ring  are 
alternately  vividly  colored  (green)  and  alternately  dark  (yellowish,  due  to 
the  dull  mercury  line) ;  i.e.,  when  there  is  interference  in  the  disk  there  is  rein- 
forcement in  the  ring,  and  vice  versa.  When  the  number  of  the  exhaustion  z 
is  high  and  the  fog-particles  therefore  large,  these  alternations  are  crowded 
so  closely  together  that  the  successive  partial  exhaustions  necessarily  skip  one 
or  more  cases;  but  for  the  smaller  particles,  i.e.,  as  far  as  exhaustion  0=15, 
the  results  are  invariably  definite.  The  curves  for  the  disk  and  inner  edge  are 
joined  by  arrows,  showing  the  successive  position  of  vivid  green  color. 

There  are  two  points  of  view  from  which  these  results  may  be  interpreted. 
We  may  lay  off  the  intensity  of  green  color  in  the  disk  and  ring  separately, 
in  which  case  maxima  of  the  one  will  coincide  with  minima  of  the  other  as 
shown  at  the  bottom  of  the  chart,  fig.  49,  i  referring  to  the  disk,  2  to  the  ring. 
Again,  the  phenomenon  may  be  regarded  as  a  series  of  contracting  rings,  which 
are  first  seen  in  a  ring  and  successively  in  the  disk.  Lines  a,  b,  and  c  have  been 
drawn  from  this  point  of  view.  The  latter  can  not,  however,  be  correct,  since 
it  frequently  happens  that  vivid  green  color  is  absent  both  from  the  ring  and 
from  the  disk,  a  condition  not  suggested  by  the  lines  a,  b,  c.  On  the  other  hand, 
the  lower  intensity  curves  of  the  figure  call  for  just  this  result  at  e,  f,  etc., 
so  that  the  former  point  of  view  is  in  accordance  with  observation.  Apart 
from  the  outer  rings,  which  are  too  large  for  observation,  we  may  even  add 
that  the  axial  colors  treated  at  the  beginning  of  the  paragraph  probably 
again  reverse  the  alternations  of  the  disk,  or  again  approach  the  case  of  the 
ring,  though  direct  observations  on  this  feature  were  not  made. 

In  fig.  50  the  same  attempt  is  made  to  represent  the  periodicity  in  disk  and 
inner  ring,  by  representing  the  apertures  of  coronas 

s  =  2Rsm8;   #  =   0  cm. 


90     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


in  terms  of  the  diameter  d  of  the  cloud  particles,  computed  from  the  successive 
exhaustions.  Here  one  may  average  the  diameter  corresponding  to  vivid  green 
color  in  the  order 

Disk,  io4d=  1.2         2.5         4-o         5-5         7-o?         8.5?  cm. 

Ring,  io4d  =        1.8         3-2         4-7         6.5         7.7?  cm. 

where  above  10*^=6.5  on.,  the  removal  by  exhaustion  is  initially  too  rapid 
to  catch  the  successive  cases  of  maxima.  If  we  also  remember  that  in  the  first 
exhaustion  the  coronas  are  filmy,  it  is  thus  a  plausible  result  of  observation 
that  the  vivid  green  disk  reappears  whenever  the  diameter  of  particles  in- 
creases by  a  quantity  less  than  1.5  X  io~4  cm.  and  greater  than  1.2  X  io~4  cm. 


If  the  prolonged  occurrence  of  vivid  color  throughout  several  exhaustions  at 
first,  and  the  unavoidable  accumulation  of  errors  in  the  successive  exhaustions 
toward  the  end  of  the  series,  be  taken  into  consideration,  it  is  not  unreason- 
able to  adjust  the  maxima  of  disk  color  for  diameters  of  fog-particles  in- 
creasing in  multiples  of  1.3 Xio-4  cm.  as  follows: 


No. 


6.5 
5 


7.8  cm. 
6 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      91 


It  is  certain,  moreover,  that  the  first  of  these  coronas  is  essentially  the  first 
in  order,  inasmuch  as  no  other  corona  can  precede  it.  It  would  follow  that 
the  reddish  coronas  first  visible  when  white  light  is  used  would  correspond  to 
particles  (X  =  63Xio-«)f 


=io-4~- 
63 


1.3  =  1.12X10-*  cm. 


or  not  below  io~4  cm.  in  diameter. 

In  a  general  way  these  conclusions  agree  with  the  datum  estimated  for  the 
axial  colors  of  coronas,  where  axial  yellow,  corresponding  to  second  green 
coronas,  was  referred  to  particles  of  the  order  of  size  io4d  =  2.2  cm.  In  both 
cases  the  assumption  is  made,  of  course,  that  all  nuclei  are  caught  in  the  exhaus- 
tion. In  fig.  51  I  have  shown  an  attempt  to  regard  the  middle  of  the  first 
ring  as  the  minimum  corresponding  to  the  disk,  as  the  intensities  alternate. 
In  the  graph  the  abscissas  denote  the  diameter  of  fog-particle  in  io~4  cm., 
computed  from  the  successive  exhaustions.  The  ordinates  of  the  curves  a 


xX 


'dm-* 


u, 


£3456 
FIG.  51. 


then  indicate  the  diameter  of  particle  if  the  aperture  s  of  the  middle  of  the 
first  ring  is  taken  in  the  equation  d=2RC\/s;  whereas  the  ordinates  of  b  show 
the  diameter  of  particle,  when  the  aperture  is  measured  as  far  as  the  middle 
of  the  dark  zone  between  the  disk  and  first  ring;  i.e.,  midway  between  the 
edge  of  the  disk  and  inner  edge  of  the  first  ring.  Clearly  the  curve  b  is  prefera- 
ble. As  far  as  particles  of  the  order  of  size  d  —  3  X  io~4  both  methods  (curve  6) 
agree  within  the  errors  of  observation;  but  for  particles  less  in  diameter,  the 
data  of  the  optical  method  are  increasingly  too  large  as  the  particles  become 
smaller. 


92     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


If  partially  monochromatic  light,  like  that  of  the  arc  lamp  filtering  through 
ruby  glass,  is  used,  the  graph  obtained  on  successive  exhaustions  is  sinuous, 
indicating  the  alternate  illumination  of  the  disk  and  first  ring;  but  it  is  not 
sharp  and  clear,  as  in  the  case  of  strictly  monochromatic  light. 

PART  II.— DISPLACEMENT  INTERFEROMETRY  WITH  FILM  GRATING. 

52.  Introductory. — The  transparent  plate  grating  is  expensive  and  rela- 
tively unavailable,  owing  to  the  fact  that  the  glass  soon  injures  the  diamond 
edge  of  the  ruling  machine.  It  is  therefore  desirable  to  attempt  to  replace  it 
by  the  film  grating,  now  so  admirably  made  by  Mr.  Ives  and  others.  Some 
time  ago  I  showed  that  this  is  quite  possible,  though  the  ellipses  obtained  were 
not  comparable  in  definition  with  those  of  the  ruled  glass  plate.  The  following 
paragraph  is  an  attempt  to  improve  the  former  method. 

The  arms  of  the  interferometer  used  were  nearly  1 50  cm.  long.  Hence  the 
tests  made  are  throughout  severe.  It  is  shown  in  the  following  paragraphs 
that  though  any  kind  of  film  grating  may  be  used,  the  particular  type  in  which 
the  smooth  side  of  the  film  is  cemented  to  plate  glass  while  the  ruled  side  is 
exposed  is  nearly  as  serviceable  as  the  ruled  plate  grating. 


i     Z     I 


4     6 


FIG.  52 


FIG.  53. 


53.  Films  between  glass  plates.— This  is  the  usual  form  of  the  grating  in 
the  market,  the  unruled  side  of  the  film  (15,000  lines  to  the  inch)  having  been 
attached  with  Canada  balsam  to  one  of  the  plates,  while  a  thin  film  of  air 
separates  the  ruled  side  from  the  other  plate.  If  this  grating  is  used  in 
the  interferometer  there  must  be  three  reflections  on  one  side  of  the  grating 
and  three  on  the  other,  supposing  that  the  ruled  face  of  the  film  and  the  glass 
face  are  practically  continuous. 

Fig.  52  represents  the  case  when  the  rays  return  upon  themselves.  The 
white  rays  L  from  the  collimator  are  separated  into  the  groups  1,2,3,  reflected 
respectively  from  the  front,  the  rear,  and  the  intermediate  faces  of  the  grating 

r  (gg  being  the  film),  toward  the  opaque  mirror  N,  whence  they  return  into 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      93 

*',  *',  3',  to  enter  the  telescope.  Similarly  the  transmitted  rays  4,  5,  6,  after 
reflection  from  the  opaque  mirror  M,  are  separated  into  the  three  groups 
4',  5',  6'  reflected  from  the  front,  the  intermediate,  and  the  rear  faces  of  the 
grating  towards  the  telescope.  It  follows,  therefore,  that  the  ray  6',  reflected 
from  the  rear  face  only,  can  have  no  spectrum,  since  it  does  not  pass  through 
the  film  g.  The  others,  i,  2,  3,  4,  5,  do  pass  through  the  film,  and  when  super- 
posed in  pairs  must  give  rise  to  elliptic  interference  in  the  superposition  of 
the  spectra.  The  figure  shows  the  refracted  rays  R  and  the  diffracted  rays  D, 
just  before  entering  the  telescope. 

Since  the  glass  is  ordinary  plate,  it  is  almost  invariably  slightly  wedge- 
shaped  and  consequently  direct  rays  will  be  seen  in  the  telescope  as  six  parallel 
lines.  These  are  shown  in  fig.  53  in  the  form  actually  observed  with  the  present 
grating.  The  rays  i,  4  from  the  front,  2,  6  from  the  rear  of  the  plates,  are 
dazzlingly  white,  whereas  3  and  5  from  the  film  within  are  respectively 
brownish-yellow  and  yellowish.  The  latter  are  thus  easily  distinguished,  but 
are  otherwise  adequately  white  and  perfectly  sharp.  It  follows  also  that  the 
air-film  is  plane  parallel,  otherwise  there  would  be  two  images  from  this  re- 
gion; that  it  does  exist,  however,  is  shown  by  the  reticulated  interferences 
below,  §§55  and  56. 

If  the  group  of  rays  i,  2,  3,  is  separated,  all  but  one  may  be  screened  off, 
and  this  is  frequently  possible.  It  is  much  more  difficult,  however,  to  screen 
off  the  rays  4',  5',  6'  individually  as  a  rule,  though  if  the  angle  of  incidence  is 
large  (45°  as  compared  with  15°  in  this  paper)  this  may  also  be  done.  No.  6' 
is  of  no  consequence,  since  it  does  not  appear  in  the  spectrum;  No.  4'  may 
be  removed  by  blacking  a  vertical  line  at  the  point  a  in  the  diagram;  or  a 
narrow  screen  (vertical  rod)  may  be  placed  at  the  proper  position  just  behind 
the  grating.  The  object  in  view  is  to  admit  only  the  two  spectra  which  are 
to  interfere  in  the  telescope,  as  this  sharpens  the  black  lines  enormously. 

It  is  for  this  reason  that  it  is  undesirable  to  have  the  rays  return  upon 
themselves;  i.e.,  they  should  not  be  reflected  from  M  and  TV  quite  normally. 
In  such  a  case,  as  will  presently  appear  more  clearly,  the  blotting  out  of  indi- 
vidual rays  or  spectra  is  more  easily  accomplished. 

In  order  that  the  slit  images  may  be  superposed  horizontally  and  vertically, 
a  fine  wire  is  drawn  across  the  slit  of  the  collimator,  the  wire  being  imaged  by 
the  black  spot  on  each  line,  as  suggested  in  fig.  53.  The  adjustment  screws 
(horizontal  and  vertical  axes  of  rotation)  on  the  mirror  then  enable  the  observer 
to  bring  any  two  slit  images  and  the  corresponding  spots  into  coincidence. 
This  adjustment  must  be  made  accurately,  if  the  interferences  are  to  be  seen 
in  the  spectra  in  the  field  of  the  other  telescope. 

54.  Continued.  The  groups  1  +4,  1  +5. — The  character  of  these  interfer- 
ences, in  which  the  rays  i  and  4  or  i  and  5  are  superposed  horizontally  and 
vertically,  is  shown  in  figs.  54  and  5  5 .  The  rays  do  not  return  upon  themselves, 
L  being  the  impinging  white  vertical  sheet  of  light,  M  and  N  the  opaque 
mirrors,  g  the  film  inclosed  between  the  plates  GG'  which  are  equally  thick. 


94     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

In  the  case  1 +  4,  the  component  rays  pass  through  the  grating,  one  and  three 
times,  respectively.  The  corresponding  path  difference  may  be  called  2  5.  The 
micrometer  position  of  the  mirror  M  was  at  the  arbitrary  mark  0.525  cm. 
The  interferences  were  very  fine  hair-lines,  evidencing  a  very  distant  center. 
They  had  the  usual  tendency  of  revolving  from  vertical  to  horizontal  when 
they  are  thickest,  into  vertical  again,  when  M  moves;  i.e.,  enormous  concentric 
equidistant  circles  are  moving  through  the  field  horizontally,  the  center  being 
invisible.  The  data  correspond  roughly  to  the  horizontal  at  the  sodium  line. 

Case  1+5  showed  the  same  phenomenon  of  fine  revolving  hair-lines,  seen 
to  be  circles  when  horizontal.  Their  micrometer  position,  however,  was  1.375 
cm.,  roughly,  so  that  the  displacement  of  M  of  0.75  cm.  has  intervened.  The 
path  difference  is  8. 

Neither  of  these  cases  is  of  value  unless  the  centers  happen  to  be  close  at 
hand.  This  is  a  matter  of  chance  residing  in  the  wedge  shape  of  the  plate. 


FIGS.  54  TO  59. 

55.  Continued.  The  groups  2+4,  2+5.— Both  of  these  interference  pat- 
terns are  interesting,  as  the  lines  are  strong  or  centered  ellipses.  The  nature 
of  the  interferences  is  given  in  figs.  56  and  57  with  the  same  notation. 

In  case  2+4,  both  component  rays  pass  through  the  glass  twice.  The  path 
difference  is  therefore  6  =  0,  and  the  micrometer  position  of  M ,  2.025  cin-'  an 
advance  of  0.75,  as  before.  The  system  is  self -compensated.  The  fine,  very 
strong,  revolving  lines,  circular  when  horizontal,  are  reticulated  (the  center, 
however,  not  being  in  the  field),  showing  a  second  set  of  interferences  not 
nearly  so  strong,  crossing  the  former  set  usually  at  about  45°.  The  fainter 
lines  may  be  obtained  by  slightly  revolving  the  mirror  M  about  the  vertical 
axis  by  aid  of  the  adjustment  screws.  These  reticulated  forms  occur  only  for 
the  compensated  adjustment.  They  are  due  to  the  two  reflections  possible 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      95 

on  the  two  sides  of  the  very  thin  air  film.  Probably  the  reflections  from  the 
film  to  glass  and  return  constitute  the  weak  lines,  the  strong  lines  not  being 
reflected  at  the  film,  but  directly  diffracted  and  transmitted.  The  reticulation 
does  not  interfere  with  the  sharpness  of  the  strong  lines. 

The  case  2+5  appears  for  the  micrometer  reading  at  the  mirror  M,  2.775 
cm.,  a  final  advance  of  0.75  cm.  Since  the  group  2  passes  through  the  glass 
twice  and  the  group  5  but  once,  the  path  difference  is  now  —5.  The  path 
difference  —26  will  not  occur,  as  the  ray  6  has  no  spectrum. 

The  present  superposition  2+5  gives  the  only  ellipses  obtainable  with  their 
centers  in  the  field.  They  are  sharp  in  line,  but  usually  somewhat  weaker 
than  desirable  in  practice,  unless  the  other  spectra  are  blotted  out  as  suggested 
in  §  53.  In  such  a  case  they  become  quite  as  available  as  the  ellipses  from  a 
plate-glass  grating. 

56.  Continued.    The  groups  3+4  and  3+5.— Both  of  these  ( see  figs.  58 
and  59)  are  interesting  in  case  of  the  given  grating. 

The  case  3+4  appears  when  the  micrometer  reading  at  M  is  1.275  cm., 
seeing  that  whereas  one  component  ray  passes  through  the  glass  3/2  times, 
the  other  passes  but  1/2  times,  the  rays  starting  in  the  middle.  The  path 
difference  is  thus  again  5,  but  the  interferences  are  thoroughly  different,  natu- 
rally, since  the  front  glass  plate  is  in  question.  They  are  very  coarse,  strong, 
revolving  lines,  curved  when  horizontal.  The  center  is  not  far  distant,  but 
outside  of  the  field.  What  is  very  striking  is  the  rapidity  of  their  revolution. 
They  pass,  almost  at  once,  from  vertical  to  horizontal  and  back  to  vertical 
again,  and  are  thus  a  sensitive  criterion  for  the  position  of  the  mirror  M  on 
the  micrometer. 

Finally,  the  case  3+5,  since  each  of  the  component  rays  passes  through  the 
glass  once,  reproduces  the  compensated  position  5  =  o,  with  the  micrometer, 
M  at  2.025.  The  fringes  are-the  same  strong  reticulated  set  described  above. 
Both  reflections  of  the  component  rays  take  place  at  the  same  face. 

57.  Centers  of  ellipses. — To  bring  the  centers  of  the  ellipses  into  the  field 
if  the  former  are  near  at  hand,  the  observing  telescope  may  be  raised  or 
lowered,  provided  of  course  the  vertical  extent  of  the  entering  pencil  of  light 
is  larger  than  the  diameter  of  the  objective. 

If  the  center  is  far  removed,  however,  the  grating  and  simultaneously  the 
mirror  N  must  be  correspondingly  inclined,  so  as  to  bring  the  entering  pencil 
of  light  from  N  back  again  into  coincidence  with  the  pencil  from  M.  Thus 
the  coarse  fringes  with  M  at  1.25  cm.,  when  thus  explored,  are  found  to  be 
very  eccentric  ellipses  with  the  long  axis  vertical,  which  accounts  for  the  rapid 
rotation  mentioned  above.  In  other  words,  when  the  spot  of  light  on  N  is 
near  the  top  (grating  and  N  being  reciprocally  inclined),  the  field  intersects 
the  top  end  of  the  major  axes  and  the  curvatures  may  be  seen.  When  the  spot 
on  N  is  near  its  bottom,  the  lower  ends  of  the  major  axes  may  be  seen.  Finally, 
when  the  spot  on  N  has  an  intermediate  position,  the  lines  no  longer  rotate, 


96     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

but  pass  from  fine,  through  coarse,  into  fine  again,  when  M  moves,  indicating 
that  the  horizontal  minor  axes  of  the  ellipses  now  occupy  the  middle  of  the 
field  of  the  telescope.  Scarcely  any  curvature  of  the  lines  can  now  be  seen. 
In  the  first  position  (top  of  ellipses  visible),  o.oooi  cm.  produces  very  visible 
change  of  inclination  of  the  nearly  horizontal  lines. 

With  the  very  fine  lines  even  this  adjustment  fails  to  bring  the  center  of 
ellipses  into  the  field.  Thus,  in  the  compensated  case,  2+4,  with  M  at  2.025 
cm.,  the  spot  of  light  on  the  opaque  mirror  N  may  (with  simultaneous  incli- 
nation of  the  grating)  be  moved  fully  10  cm.  vertically,  without  producing 
marked  effect  on  the  interference  pattern.  The  lower  parts  of  enormous 
circles  are  intersected  by  the  field  of  the  telescope  throughout. 

58.  Film  or  ruling  on  one  side  of  the  glass  plate.    Ruled  grating.— If  the 

film  is  on  one  side  only,  the  case  of  the  plate-glass  grating  is  reproduced.  From 
the  absence  of  so  many  superposed  spectra,  the  results  should  therefore  be 
better. 

The  plate-glass  grating,  10,000  lines  to  the  inch,  mounted  in  the  interferom- 
eter with  its  ruled  face  toward  the  collimator,  shows  but  two  lines  for  the 
reflection  from  each  of  the  opaque  mirrors,  of  which  three  only  give  rise  to 
spectra,  the  beam  of  the  fourth  being  reflected  from  the  rear  face  and  not 
passing  through  the  grating,  after  reflection.  But  this  case  may  be  reached 
by  reversing  the  grating  (ruled  face  toward  the  telescope),  under  which  con- 
ditions all  the  rays  have  spectra.  The  micrometer  positions  for  these  inter- 
ferences were  as  follows: 

M  at  2.95  cm.  Ruled  face  in  front.  Reflection  from  the  front  face.  Path 
difference  5.  Reflection  from  the  rear  face  may  be  screened  off  at  the  opaque 
mirror  N.  Fine,  sharp,  solitary  ellipses,  very  black  lines  on  an  even,  brilliantly 
colored  ground.  No  stationary  interferences  from  the  front  and  rear  faces  of 
the  grating. 

M  at  1.90  cm.  Ruled  face  front  or  rear.  Reflection  from  front  and  rear 
faces  conjointly.  Compensated  position.  Path  difference  5  =  o.  Fine  revolv- 
ing hair-lines  belonging  to  a  remote  center. 

M  at  0.85  cm.  Ruled  face  to  the  rear.  Path  difference  -S.  Fine  ellipses, 
but  complicated  by  stationary  interferences  and  superposed  spectra,  which 
can  not  be  easily  screened  off.  The  first  position  is  here  also  available. 

The  position  (2.95  cm.)  with  ruled  face  toward  the  collimator,  is  thus  the 
practical  case,  since  it  is  sharpest  and  without  complications. 

The  micrometer  displacement  AW,  which  should  correspond  to  the  extreme 
positions  of  the  grating  (ruled  face  forward  and  ruled  face  rearward),  if  e  is 
its  thickness,  I  and  R  the  angles  of  incidence  and  refraction,  M  the  index  of 
refraction,  is 

AA/"  =  2^  cos/? 
Here  roughly  7=i5°;  tf=9.7o.M=I53;<,  =  o67  ^     Hence 

AAf=  2X0.67X1. 53X0.986  =  2.02  cm. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.      97 

The  difference  (observed  value  2.1  cm.)  is  due  to  the  necessary  reversal  of 
the  grating,  which  requires  considerable  readjustment.  Using  the  third  posi- 
tion, the  observed  A/V  was  more  closely  2.059  cm.  But  here  again  two  separate 
pairs  of  slit  images  must  be  placed  in  coincidence,  which  calls  for  a  double 
rotation  of  the  mirror  M. 

59.  Continued.    Film  grating  not  cemented  to  glass.— To  test  this  case 
for  the  film  grating,  a  number  of  fine  specimens  kindly  made  for  me  by  Mr. 
Ives  were  at  hand.    The  film  of  these  was  not  attached  continuously  to  the 
glass  by  a  layer  of  Canada  balsam.     Nevertheless  the  film  was  perfectly 
smooth  and  reflected  well.    It  had  about  15,000  lines  to  the  inch,  which  is 
rather  too  great  a  number  for  work  of  this  kind,  as  the  ellipses  have  their  long 
major  axes  horizontal.    Accuracy  is  enhanced  if  the  major  axes  are  vertical. 
Furthermore,  the  spectrum  toward  the  right  of  the  refracted  ray  is  too  far 
distant  from  the  micrometer  for  the  easy  manipulation  of  the  latter.    Hence 
the  spectra  toward  the  left  must  be  used,  if  available. 

The  refracted  slit  images  are  shown  in  fig.  60,  when  the  ruled  face  is  toward 
the  rear.  There  are  but  two  lines  visible  from  the  opaque  mirror  M  and  three 
from  the  opaque  mirror  N,  the  fainter,  yellow  and  less  even  line,  No.  3,  coming 
from  the  film.  To  obtain  the  ellipses  which  are  here  alone  sought,  lines  2  and 
3  reflected  from  the  same  face  are  put  in  full  coincidence. 
The  micrometer  positions  were  approximately  as  follows: 

M  at  2.9  cm.     Film  toward  the  collimator.    Good  flat 
ellipses,  but  not  very  strong.    Path  difference  5. 

M  at  2.0  cm.    Compensated  position.    Path  difference 
5  =  o.     Fine  hair-lines,  revolving. 

M  at  i.i  cm.     Film  toward  the  telescope.      Path  dif- 
ference —  5.    Ellipses  resembling  the  first  case. 

The    stationary  interferences  were  present,   but  not 
objectionable.     The    spectrum,  showing  a  very  bright       ^  /y 

sodium  line,  is  brilliant.  pIG  fio 

A  large  number  of  other  film  gratings  of  the  same  kind 
were  tested,  but  with  no  further  results.  Sometimes  there  are  four  slit  images, 
at  other  times  six,  depending  upon  the  adjustment  and  shape  of  the  films.  The 
tests  were  somewhat  severe,  as  the  arms  of  the  interferometer  were  nearly  150 
cm.  long;  but  it  is  clear  that  in  order  to  get  the  best  results  the  film  should  be 
continuous  with  the  glass,  being  cemented  with  Canada  balsam  on  the  unruled 
side,  while  the  ruled  side  is  outermost.  In  this  case  the  plate  grating  is  almost 
reproduced. 

60.  Single  plate,  film  grating.— It  has  been  stated  that  the  best  results 
would  be  obtained  with  a  film  grating  cemented  to  plate  glass  on  the  smooth 
side,  with  the  ruled  side  exposed.    Having  failed  to  make  one  adequately 
plane  myself,  I  was  fortunate  in  securing  a  sample  through  the  courtesy  of 
Mr.  Ives.    The  number  of  lines  to  the  inch,  14,438,  was  rather  in  excess  of 


98     EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

the  number  desirable  for  interferometry,  as  the  ellipses  when  the  number  of 
lines  exceeds  10,000  to  the  inch  are  liable  to  be  too  broad  for  accurate  measure- 
ment. Otherwise  the  grating  sufficed  the  required  purposes  admirably.  The 
great  advantage  in  adjustment  in  case  of  such  a  grating  is  the  presence  of 
only  one  strong  reflection,  namely,  that  from  the  uncovered  face  of  the  glass. 
The  side  covered  by  the  grating  reflects  but  very  feebly.  Consequently  the 
grating  is  to  be  mounted  with  the  rear  side  (smooth  glass)  toward  the  source 
of  light  and  the  celluloid  side  toward  the  eye  of  the  observer.  Hence  if  the 
two  strong  slit  images  in  the  telescope  of  the  interferometer  are  placed  in 
contact  horizontally  and  vertically,  the  ellipses  are  found  for  approximately 
equal  distances,  without  difficulty.  They  are  centered,  since  both  reflections 
occur  at  the  same  position  of  the  same  face.  It  is  also  possible  to  obtain  faint 
reflections  from  the  grating  face,  the  slit  image  being  usually  deep  blue  in 
color.  In  spite  of  this,  however,  the  spectrum  (as  the  ray  does  not  again  pass 
through  glass)  is  strong.  The  interferences  are  rarely  centered,  as  the  two 
reflections  contain  the  angle  of  the  faces  of  the  glass  plate  between  them. 
They  consist  of  lines  coarsening  and  rotating  180°,  as  the  vertical  projection 
of  the  distant  center  is  passed.  The  ellipses,  if  too  broad,  may  as  usual  be 
made  smaller  with  a  thick  compensator,  but  at  a  sacrifice  of  sensitiveness. 
With  concave  mirrors,  on  the  horizontal  pendulum  for  instance,  the  ellipses 
are  apt  to  be  small  and  round,  even  if  flat  and  coarse  with  plane  mirrors.  Thus 
with  concave  mirrors  the  interposed  thick  plate  of  glass  is  not  needed. 

PART  m.— ELLIPTIC  INTERFEROMETRY  WITH  A  NERNST  FILAMENT. 

61.  Introduction. — The  ideal  illumination  for  the  present  purposes  is  sun- 
light, inasmuch  as  the  lines  of  the  spectrum  are  always  present ;  but  it  is  not 
generally  available.  In  its  absence  the  electric  arc  does  good  service.  Here 
the  sodium  line  is  always  visible  and  of  sufficient  intensity  to  be  utilized  as  a 
landmark,  if  desirable.  It  is  preferable  to  use  the  arc  without  condensers 
and  to  place  it  several  feet  from  the  slit,  in  order  that  the  rays  may  be  nearly 
parallel  and  that  the  pencil  which  comes  out  of  the  objective  of  the  collimator 
may  be  a  nearly  vertical  sheet,  converging  toward  the  opaque  mirrors.  Such 
a  rapier-like  beam  is  more  easily  made  to  penetrate  long  tubes  and  similar 
appliances.  If  the  interferences  are  to  be  sharp,  the  lateral  extent  of  the  pencil 
must  be  as  narrow  as  practicable.  Moreover,  while  the  rays  from  the  colli- 
mator are  parallel  in  their  horizontal  projection,  they  are  not  so  in  their 
vertical  projection,  for  which  case  the  focus  may  as  a  rule  be  advantageously 
placed  (by  moving  the  electric  arc  to  or  from  the  slit)  at  the  opaque  mirrors 
of  the  interferometer. 

The  arc  lamp  has  one  great  objection,  however,  inasmuch  as  the  mobile 
arc  requires  constant  attention,  and  even  in  such  a  case  adequate  illumination 
often  fails  at  a  critical  moment.  It  is  therefore  desirable  to  find  a  more  steady 
source  of  iUumination  of  sufficient  intensity,  and  this  is  clearly  attainable  with 
the  use  of  the  Nernst  burner.  Experiments  were  therefore  made  with  this 
light  and  the  following  compact  form  of  apparatus  which  satisfies  many  pur- 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.     99 

poses  and  suggests  itself  particularly  with  a  view  to  the  photography  of  the 
interferences,  such  as  may  be  needed  in  the  next  section. 

62.  The  Nernst  burner.— In  fig.  61  (horizontal  projection),  N  is  the  usual 
type  of  Nernst  burner  with  electromagnetic  base,  the  filament  being  at  a.  It 
is  inclosed  in  the  rectangular  case  A,  of  blackened  tin-plate,  judiciously  pro- 
vided with  holes  for  ventilation,  so  as  not  to  allow  the  appreciable  escape  of 
light  from  the  case,  except  in  front.  The  front  is  a  micrometer  slit  5,  in  ad- 
vance of  which  a  short  end  of  flanged  tubing  c  is  adjustable,  so  as  to  cut  off 
the  excess  of  light  spreading  up  and  down.  The  plate  holding  the  slit  5  may 
be  removed  by  sliding  it  up  on  guides.  Similarly  the  slit  5  and  the  tube  c 
may  be  removed  by  sliding  the  latter  horizontally  on  this  plate.  O  is  the  lens 
of  the  colHmator,  aO  being  (nearly)  its  principal  focal  distance.  This  is  found 
by  obtaining  the  sharp  image  of  the  slit,  reflected  by  the  opaque  mirror  N 
on  the  jaws  of  S. 


FIG.  61. 


In  order  that  the  interferences  may  be  sharp,  the  beam  of  light  falling  on 
the  grating  must  be  of  slight  extent  (>6  inch)  laterally.  For  this  purpose  a 
screen  with  an  appropriately  wide  vertical  slit  is  placed  either  at  F  or  on  the 
grating  table  at  b.  When  this  is  done  two  slit  images  may  usually  be  seen  at 
the  mirror  N  and  the  brownish  one  is  screened  off  there.  To  obtain  the  soli- 
tary ellipses,  the  ruled  side  g  of  the  grating  should  face  the  source  of  light. 
As  the  grating  is  of  ordinary  glass  plate  and  therefore  wedge-shaped,  the  top 
and  bottom  of  the  grating  should  be  selected  so  that  the  wedge  and  the  thick- 
ness effect  act  in  concert,  to  separate  the  two  slit  images  at  N,  referred  to. 
In  this  case  the  undesirable  image  may  be  more  easily  screened  off. 

63.  Remarks. — It  was  my  expectation  that  the  Nernst  filament  might 
itself  be  used  as  a  slit  without  further  appliances  than  the  screen  tube  c.  But 
this  is  not  adequately  the  case,  as  the  filament  is  a  little  too  thick.  Without 
the  slit  and  the  tube  only,  the  ellipses  are  just  suggested.  Possibly  if  the 
white  porcelain  surface  of  the  Nernst  burner  were  black  instead  of  white 
porcelain,  clearness  would  be  enhanced.  But  the  ellipses  would  not  be  useful 
for  measurement.  Without  the  slit,  but  with  the  slotted  screen  T  or  6,  the 
ellipses  are  strong  but  somewhat  washed,  so  that  the  fine  lines  to  right  and 
left  soon  vanish.  The  rings  could  actually  be  used  for  measurement,  for 


100  EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

the  centers  are  well  indicated  and  the  motion  of  rings  adequately  clear.  With 
the  slit  and  tube,  or  slit  tube  and  screen,  the  ellipses  become  sharp  and  the 
fine  lines  indefinitely  visible.  The  slit  need  not  be  very  fine,  but  as  it  is  finer 
the  velvety  black  lines  on  the  colored  spectrum  become  more  marked.  The 
interference  pattern  is  now  quite  as  good  as  with  the  arc  lamp. 

Naturally  when  the  filament  is  so  near  the  slit,  the  rays  on  leaving  the 
collimator  diverge  strongly  in  the  vertical  plane.  Hence  the  illuminated  parts 
of  the  mirrors  M  and  N  may  be  2  or  3  inches  long.  If  these  mirrors  are  of 
ordinary  plate  glass  it  is  not  liable  to  be  adequately  perfect  over  the  whole 
length,  and  the  ellipses  will  be  imperfect  in  form.  But  this  is  not  a  serious 
disadvantage. 

At  wider  ranges  (g  to  M  and  N  i  to  2  meters),  the  arrangement  is  not  very 
satisfactory  for  photography,  because  the  light  passing  through  the  telescope, 
unless  the  objective  is  very  large  (a  ^-inch  objective  was  used),  is  only  a 
small  part  of  that  passing  through  the  slit.  Hence  the  light  camera  at  the 
end  of  the  telescope  is  insufficiently  illuminated.  For  photographic  purposes 
it  would  then  seem  to  be  better  to  place  the  Nernst  filament  at  a  distance 
from  the  slit  and  to  use  a  condenser;  but  I  was  unable  to  obtain  marked  ad- 
vantages in  this  way,  while  the  condenser  is  an  annoyance.  Hence  for  photo- 
graphic purposes  it  is  better  to  replace  the  plane  mirrors  M  and  N  by  identical 
concave  mirrors  in  which  the  light  is  appropriately  condensed.  This  is  done 
in  the  inclination  apparatus  in  Chapter  I,  §  21,  and  further  reference  has  been 
made  there.  In  any  case,  however,  greater  steadiness  and  freedom  from 
tremor  than  the  laboratory  affords  would  be  desirable  for  photography,  and 
though  it  is  not  difficult  to  obtain  families  of  ellipses  in  the  way  given  on  the 
ground-glass  screen  of  the  camera,  few  experiments  in  actual  photography 
were  made. 

The  spectrum  of  the  Nernst  filament  is  free  from  the  Fraunhofer  lines.  It 
is,  however,  easy  to  obtain  the  reversed  D  lines,  by  using  an  ordinary  sodium 
flame  placed  either  in  front  of  the  slit  or  (contrary  to  expectations)  even  be- 
hind it  within  the  collimator.  One  would  have  expected  the  latter  method 
to  interfere  with  the  definition,  but  it  does  not  seem  to  do  so.  When  the 
sodium  lines  have  once  been  indicated,  the  cross-hairs  of  the  telescope  may 
be  placed  in  coincidence  with  them  and  the  desirable  fiducial  lines  of  the  spec- 
trum thus  obtained. 

PART  IV.-SCATTERING   IN  THE  CASE  OF  REGULAR  REFLECTION  FROM  A 
TRANSPARENT  GRATING,  AN  ANALOGY  TO  THE  REFLECTION 

OF  X-RAYS  FROM  CRYSTALS. 

64.  The  phenomenon.— No  doubt  the  following  phenomenon  has  been 
noticed  before,  but  I  have  seen  no  description  of  it.  If  a  vertical  sheet  of 
white  light  L,  from  a  collimator,  is  reflected  from  the  two  faces  of  a  plate- 
glass  grating,  having  about  10,000  or  more  lines  to  the  inch,  g  being  the  ruled 
face,  the  two  beams  6  and  y  going  to  the  opaque  mirror  N  are  respectively 
vividly  blue  and  brownish  yellow.  In  other  words,  more  blue  light  is  regu- 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.   101 


larly  reflected  from  the  ruled  surface  than  is  transmitted,  and  more  reddish 
light  transmitted  than  is  reflected.  Since  the  plate  grating  is  not  quite  plane 
parallel,  two  of  the  four  rays,  b'  and  y',  are  seen  in  the  same  colors  in  the 
telescope.  This  is  a  great  convenience  in  adjusting  the  displacement  inter- 
ferometer, where  the  spectra  from  b  alone  are  wanted,  and  the  y  ray  may  be 
screened  off  at  N,  while  the  other  y1  has  no  spectrum. 

The  transmitted  rays  t  after  reflection  show  very  little  difference,  the  one 
reflected  at  g  being  perhaps  slightly  yellowish  as  compared  with  the  other. 

The  spectra  from  6  and  y,  if  compared  one 
above  the  other,  are  practically  identical. 
The  difference  is  not  sufficiently  marked  to 
be  discerned  by  the  eye.  Multiple  reflection 
from  the  two  faces  gave  no  further  results. 

Finally,  to  be  colored  blue,  the  beam  must 
be  reflected  from  the  air  side  and  not  from 
the  glass  side,  where  but  little  appreciable 
effect  is  produced.  If  the  grating  is  turned 
1 80°,  both  the  6  and  y  rays  are  nearly  white, 
while  the  t  rays  now  correspond  to  the  b  and 
y  rays  and  are  vividly  colored. 

Outside  the  ruled  surface  and  with  any  or- 
dinary unruled  plate  of  glass,  all  images  are  XT' 
of  course  white.  I  mention  this  merely  since  c4{  >» 
one  might  suppose  the  absorption  or  color  of 
the  glass  to  have  something  to  do  with  the 
experiment.  The  film  grating,  where  sharp  reflection  takes  place  from  the  glass 
and  not  appreciably  from  the  film,  does  not  ordinarily  show  the  phenomenon; 
but  in  case  of  the  single-plate  film  grating  of  paragraph  60,  it  is  astonishingly 
strong  in  the  refracted  slit  images  seen  in  the  telescope.  These  are,  in  fact, 
azure  blue  when  coming  from  the  mirror  N  and  reflected  from  the  front  side 
(toward  the  lamp)  of  the  film;  deep  brown  when  reflected  from  the  rear  side, 
after  having  passed  through  the  film.  The  two  images  may  be  superposed  by 
rotating  M  with  the  production  of  nearly  white  light.  Moreover,  the  marginal 
light  (otherwise  identical  but  not  passing  through  the  film)  is  white.  The 
images  in  question  are  sharp,  but  it  is  possible  that  the  material  of  the  film 
may  somewhat  contribute  to  the  color. 

65.  Explanation.— Scattering  is  usually  and  perhaps  essentially  associated 
with  diffuse  reflection.  The  present  phenomenon,  however,  is  strictly  regular 
reflection;  i.e.,  there  is  a  wave-front,  for  the  blue  and  yellow  slit  images  are 
absolutely  sharp  in  the  telescope.  This  is  the  interesting  feature  of  the  phe- 
nomenon, which  associates  it  at  once  with  the  recent  famous  discovery  of 
Friedrich,  Knipping,  and  Laue  relative  to  the  reflection  of  X-rays  from  the 
molecular  planes  of  crystals,  and  it  is  for  this  reason  that  I  call  attention 
to  it. 


FIG.  62. 


102  EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

In  case  of  the  grating  the  sources  of  scattered  light-waves  are  not  only 
identical  as  to  phase,  but  these  sources  are  at  the  same  time  equidistant. 
Hence  collectively  they  must  determine  a  wave-front  of  somewhat  inferior 
intensity  but  otherwise  identical  with  the  wave-front  of  normally  reflected 
or  diffracted  light;  i.e.,  the  wave-fronts  of  regularly  reflected  and  scattered 
light  are  superposed. 

Moreover,  if  the  grating  is  turned  in  azimuth  even  as  much  as  45°  on  either 
side  of  the  impinging  beam  (after  which  the  many  reflections  and  diffractions 
seriously  overlap)  the  blue  and  brown  colorations  are  distinctly  intensified. 
This  also  is  in  accordance  with  anticipations,  for  the  number  of  lines  which 
are  comprehended  within  the  lateral  extent  5  of  the  narrow  beam  L  as  the 
angle  of  incidence  *  is  varied,  increases  as  s  sec  i;  whereas  the  lateral  extent 
of  the  reflected  beam  is  no  larger  than  that  of  the  impinging  beam.  Hence 
there  should  be  increased  intensity  of  scattered  light  in  the  ratio  of  sec  i 
or  increasing  markedly  with  *  from  i  for  i  =  o°,  to  oo  for  *  =  90°.  In  other  words, 
the  scattering  lines  of  the  grating  are  virtually  more  densely  disseminated 
when  *  increases. 

For  the  light  reflected  from  the  inside  of  the  glass  plate  the  evidence  to  be 
obtained  from  color  in  case  of  the  ruled  grating  is  too  vague  to  admit  of  definite 
statements.  I  have  not,  therefore,  attempted  it. 

66.  A  further  analogy  to  the  reflection  of  X-rays.— With  regard  to  the 
recent  experiments  (I.e.)  on  the  reflection  of  X-rays  from  crystals,  it  may 
further  be  interesting  to  recall  my  experiments  (Phil.  Mag.,  xxu,  p.  12 1, 
1911)  on  the  interferences  produced  by  two  identical  but  separate  halves  of  a 
reflecting  grating,  with  the  rulings  parallel  and  originally  in  the  same  plane. 
The  interferences  observed  are  brought  out  by  moving  one  of  the  half  gratings 
micrometrically  parallel  to  itself,  to  the  front  or  to  the  rear  of  the  other  half, 
and  are  here  necessarily  linear  and  parallel  to  the  rulings.  If  i  (angle  of  inci- 
dence) =6  (angle  of  diffraction)  and  d  is  the  normal  distance  apart  of  the  grat- 
ings, the  same  equation  n\=zd  cos  6  holds.  In  other  words,  two  identical 
spectra  originating  in  parallel  planes,  at  a  distance  apart  commensurate  with 
the  wave-length  of  light,  are  superimposed  throughout  their  extent  and  pro- 
duce interferences.  I  pointed  out  the  bearing  of  this  phenomenon  on  the 
theory  of  the  coronas  of  cloudy  condensation  (I.  c.,  p.  129),  where  the  compound 
diffraction  spectra,  due  to  successive,  parallel,  equidistant  layers  of  fog- 
particles  (a  sort  of  space  lattice),  are  superimposed  and  interfere  in  a  manner 
evidenced  by  the  disk  colors  of  coronas. 

In  the  actual  case  of  distribution,  however,  the  fog-particles  (as  I  also 
pointed  out)  are  too  far  apart  to  admit  of  the  immediate  application  of  the 
direct  theory  in  question.  Some  extension  of  this  point  of  view  must  therefore 
be  forthcoming  if  the  experiment  with  halved  gratings  one  behind  the  other 
is  to  be  reconciled  with  the  circumstances  under  which  coronal  phenomena 
appear. 


CHAPTER  V. 


DISPLACEMENT  INTERFEROMETRY  APPLIED  TO  THE  QUADRANT 
ELECTROMETER. 

67.  Apparatus. — In  an  earlier  report  experiments  were  given  showing  the 
adaptation  of  the  quadrant  electrometer  for  the  measurement  of  very  small 
potential  differences,  when  the  needle  is  provided  with  two  symmetrical,  light, 
plane  mirrors,  in  parallel.  The  excursions  of  the  needle  may  be  read  off,  for 
small  angular  deviation,  on  the  displacement  interferometer.  If  5 = AN  is  the 
displacement  of  the  mirror  of  the  micrometer  of  this  instrument,  and  i  the 
angle  of  incidence  of  the  ray  impinging  on  either  of  the  small  parallel  mirrors 
on  the  needle, 

5  =  20  cos  *  —dS/di=  20  sin  t 

where  a  is  the  normal  distance  apart  of  the  parallel  mirrors.  If  degrees  of  arc 
are  used  the  ratio  is  0.035  a  SU1  *  an(i  *  is  usually  about  45°.  It  should  be  pos- 
sible with  such  an  arrangement  to  obtain  a  sensitiveness  of  a  few  millionths 
volts  per  vanishing  interference  ring,  and  the  following  paper  is  a  further 
attempt  to  reach  this  result,  practically. 

The  main,  if  not  insuperable,  difficulty  encountered  in  such  an  apparatus 
is  the  continual  and  often  irregular  drift  of  the  needle,  when  the  condition  of 
rest  is  so  sharply  determined.  A  special  environment,  without  city  tremors 
and  at  constant  temperature,  seems  to  be  the  only  means  of  obviating  these 
annoyances. 

The  apparatus  used  is  shown  in  fig.  63  in  vertical  section.  AA  is  the  per- 
forated base  of  a  massive  brass  plate,  i  cm.  thick,  securely  fastened  by  a  large 
clamp  to  one  arm  of  the  interferometer,  capable  of  some  rotation  around  the 
vertical  and  horizontal  axes  for  leveling  the  whole  apparatus,  etc.  To  this  the 
quadrants  a,  b  are  firmly  attached,  by  aid  of  screws  i,  j,  but  in  such  a  way  as 
to  be  quite  insulated  from  the  brass  plate,  in  view  of  the  perforated  columns 
g,  h  and  nuts  u,  v  of  hard  rubber  and  of  the  form  shown.  The  clamp-screws 
k,  I  are  in  metallic  contact  with  i,  /,  and  carry  charges  to  the  quadrants.  There 
are  about  2  inches  of  free  space  below  the  plate  A  A,  available  for  the  connec- 
tions and,  if  necessary,  for  a  liquid  damper,  w. 

The  needle  consists  of  two  8 -shaped  leaves,  c  and  d  (biplanes),  symmetri- 
cally fastened  to  the  stem  st,  on  which  the  needle  is  bifilarly  suspended  from 
silk  fibers.  The  two  small  parallel  mirrors,  e  and  /,  are  adjustably  attached 
to  a  fine  metallic  wire  at  right  angles  to  st  and  in  contact  with  d.  Each  mirror 
has  a  light  vertical  and  horizontal  axis  in  a  bit  of  cork  (not  shown).  The 
mirrors  are  first  made  parallel  by  using  sunlight  and  then  fixed  with  melted 
wax,  after  which  the  aluminum  foils  c  and  d  are  centered  in  place,  the  eyelets 
at  s  and  t  having  not  as  yet  been  bent.  Light  reaches  the  mirrors  e,  f  through 
two  corresponding  holes  cut  in  the  vertical  walls  of  the  quadrants.  The 

103 


104  EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

weight  of  such  a  needle  is  easily  kept  within  0.75  gram  and  the  air-damping  is 
quite  sufficient.  Unfortunately  its  period  is  large,  being  about  i  minute,  and  it 
is  apt  to  vibrate  as  a  pendulum.  Hence  it  is  often  convenient  to  hook  on  a  wire 
at  t,  bent  like  an  inverted  V,  with  the  free  ends  submerged  in  water  to  secure 
greater  steadiness  on  the  interferometer;  or  a  mica  vane  may  be  added,  as  at  w. 
The  bifilar  suspension,  13.5  to  20.2  cm.  long  in  the  different  experiments, 
terminating  above  in  the  hooked  brass  rod  r,  is  adjustably  fixed  in  the  brass 
cylinder  p,  which  in  turn  is  secured  in  the  hard-rubber  insulator  n,  attached  at 
right  angles  to  the  brass  standard  GG,  the  lower  end  of  which  is  screwed  to  the 

a 


FIG.  63. 

brass  plate  A  A.  This  rod  can  be  lengthened  telescopically  (not  shown)  ad- 
mitting of  different  lengths  of  bifilar  suspension.  The  hard-rubber  lever  0 
enables  the  observer  to  twist  the  bifilar.  The  charge,  from  a  Zamboni  cell 
or  the  lighting  circuit  (250  volts),  is  conveyed  to  the  needle  through  the  hard- 
rubber  insulator  at  m  and  the  clamp-screw  at  q  (which  in  turn  secures  the 
plug  p),  through  the  moistened  bifilar  wires,  as  in  Dolezalek's  apparatus;  or 
it  may  be  admitted  through  the  insulated  damper  below  t. 

Finally,  the  lower  part  of  the  case  CD  envelops  the  quadrants  more  or 
less  permanently  and  is  provided  with  wide  plate-glass  windows  for  observa- 
tion. The  upper  part  EF  of  the  case  may  be  taken  off  like  a  hat. 

68,  Observations. — Experiments  were  made  with  this  apparatus  at  con- 
siderable length,  but  they  were  not  sufficiently  definite  to  lead  to  any  quanti- 
tative statement.  Great  difficulty  was  experienced,  in  addition  to  the  drift 
of  the  needle,  in  securing  an  adjustment  of  the  mirrors  such  that  the  beam  of 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.   105 

light  might  pass  from  mirror  to  mirror  and  return  through  the  inside  of  the 
quadrants.  For  since  the  mirrors  are  quite  inclosed  in  the  latter,  the  path  of 
the  beam  of  light  can  not  easily  be  seen,  and  it  is  troublesome  to  obtain  the 
several  reflections  to  the  best  advantage.  As  the  needle  fits  the  quadrants 
with  but  one-eighth  inch  of  clear  space,  it  is  very  liable  to  be  unstable  if  the 
parts  of  it  are  but  slightly  out  of  true. 

69.  Observations,  continued.— For  this  reason  it  was  thought  preferable 
to  conduct  the  experiments  by  using  a  needle  provided  with  mirrors  attached 
on  the  outside  of  the  quadrants.    Such  a  needle  (No.  i)  is  shown  in  fig.  64 
(of  the  wedge  type)  at  cd,  the  two  8 -shaped  leaves  meeting  on  the  outside, 
in  a  horizontal  circular  arc  xy.    The  mirrors  with  the  axles  in  cork  are  shown 
at  eft  and  should  be  several  centimeters  above  the  quadrants  ab.    The  adjust- 
ment here  is  comparatively  easy,  as  the  mirrors  and  the  path  of  light  are  all 
quite  visible.    The  needle,  being  sharp-edged,  may  be 

charged  to  a  potential  of  several  hundred  volts,  with- 
out  instability.    The  period,  however,  is  still  large. 

In  the  first  series  of  experiments  the  needle  was 
charged  with  a  Zamboni  cell  to  about  150  volts,  and 
the  voltage  measured  at  the  quadrants  was  about 
0.04  volt.  The  ellipses  showed  continual  drift,  the 
needle  moving  as  if  a  force  acted  in  one  direction 
for  a  time  large  as  compared  with  the  period  of  the  {, 

needle.    The  mirrors  were  slightly  curved,  so  that  in  FlG-  64- 

place  of  ellipses  the  interference  figures  were  lemniscates.  In  spite  of  the  diffi- 
culties, the  two  series  of  experiments  show  sensitiveness  of  0.5  and  0.4  cm. 
per  volt,  respectively,  which  is  equivalent  to  about  7  X 10-*  volt  per  vanishing 
interference  ring. 

Using  the  same  needle,  the  voltage  was  now  presumably  doubled  by  using 
two  Zamboni  piles.  The  sensitiveness,  however,  not  only  was  not  enhanced, 
but  showed  a  decrease,  0.02  volt  being  measured.  In  other  experiments  the 
sensitiveness  was  successively  0.5,  0.4,  0.4  cm.  per  volt,  respectively. 

70.  Observations,  continued. — The  sensitiveness  was  now  increased  by 
using  a  new  needle  (II)  of  the  form  given  in  figs.  6$A  and  656.    The  two  8- 
shaped  leaves  or  biplanes,  c  and  d,  of  the  needle  are  parallel  and  the  circular 
edges  at  x  and  y  closed  with  parts  of  cylindrical  shells  of  aluminum  foil.    It 
is  presumable  from  the  elementary  theory  of  the  instrument  that  these  walls 
x,  y  must  contribute  essentially  to  its  sensitiveness.     In  the  present  case  the 
capsule  of  the  quadrants  (II)  was  about  10  cm.  in  diameter  and  about  2  cm. 
in  vertical  height,  within,  with  a  length  of  the  needle,  xy,  of  about  9  cm.  and 
a  distance  apart  of  the  biplanes  c  and  d  about  0.8  cm.    This  gave  about  0.5 
cm.  of  dear  space  at  the  ends  and  about  0.6  cm.  of  clear  space  above  and  below 
the  needle,  as  an  allowance  for  stability.    The  needle  swung  freely  and  was 
inserted  without  difficulty.    The  mirrors  were  about  8  cm.  apart.    To  secure 
greater  steadiness  a  water  damper  was  installed  below,  though  it  would  not 


106  EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 

otherwise  have  been  necessary.  The  drift  was  at  first  marked,  but  finally 
subsided  to  a  reasonably  small  value.  The  sensitiveness  (centimeters  of  dis- 
placement per  volt)  in  successive  groups  of  observations  was  (AF  potential 
increment,  A/V  micrometer  displacement)  AA/r/AF=i.o,  i.i,  i.i,  1.6,  0.8,  i.o, 
i.i,  1.2 ;  therefore  about  i.i  cm.  per  volt,  equivalent  to  3  X  lo"5  volt  per  vanish- 
ing interference  ring.  In  these  data  some  extraneous  oscillation  of  the  needle 
is  manifest,  which  vanishes  in  consecutive  groups  of  results. 

The  experiment  was  then  repeated  with  two  Zamboni  cells  charging  the 
needle.  This,  however,  again  actually  reduced  the  sensitiveness  to  AA/"/AF= 
0.8,  0.7,  0.6,  0.7,  0.7  cm.  per  volt,  in  successive  groups  of  observations,  a  result 
equivalent  to  4  X  iQ-6  volt  per  vanishing  ring.  The  data  remained  of  the  same 
order,  whether  the  needle  was  charged  from  above 
or  below,  so  that  it  is  inherent  in  the  theory  of  the 
instrument.  On  returning  to  the  single  Zamboni 
charging  cell,  sensitiveness  again  increased  to 
A/V/AF=  1.15,  or  to  about  26  X  io~6  volt  per  ring. 

The  Zamboni  cells  were  now  removed  and  the 
needle  charged  from  above  with  the  electric- 
lighting  circuit  at  250  volts.  To  obviate  the  effect 
of  drift,  which  is  liable  to  be  persistently  in  one 
direction,  observations  were  taken  every  1.5  min- 
utes. The  sensitiveness  in  two  groups  of  experi- 
ments of  about  5  observations  each  was  then  i.i 
and  1.4  cm.  per  volt,  or  on  the  average  24Xio-« 
volt  per  vanishing  ring.  Many  other  experiments 
were  made  with  similar  results. 

The  annoyance  of  a  drifting  needle,  which  occurs 
throughout  the  above  results  and  which  at  first 
seemed  to  have  a  definite  direction  from  the  dark 
to  the  light  side  of  the  parallel  mirrors,  was  also 


FIG.  65 


made  the  subject  of  considerable  study,  sunlight  being  used  to  avoid  the  radia- 
tions from  the  body  of  the  electric  lamp.  In  these  cases  the  displacement  of 
about  0.07  cm.  within  a  half  hour  was  usually  reversed  in  the  course  of  this  time, 
so  as  to  bring  the  needle  nearly  back  to  its  original  position.  As  the  experiments 
were  made  with  the  apparatus  uncharged,  the  only  reason  for  this  drift  seemed 
therefore  to  be  the  occurrence  of  steady  air-currents,  in  spite  of  the  protection 

>t  the  case  and  the  rapid  subsidence  of  the  pendulum  vibrations  of  the  damped 
needle.    The  attempts  made  to  obviate  these  difficulties  were  all  futile, 
i  7/;  °bserjations'  continued.-Another  biplane  needle  (III)  in  place  of  the 
last  (ngs.  65A  and  6SB)  was  now  installed.    The  blades  of  the  needle  were  1.2 
cm  £  part  and  to  give  it  stiffness  a  vertical  partition  running  symmetrically 

rom  end  to  end  was  fixed  within,  the  whole  being  of  aluminum  foil  0.002  cm. 

thick  and  the  frame,  as  before,  of  steel  wire  0.044  cm.  in  diameter.    The  weight 

e  needle  with  mirrors  adjusted  was  about  1.2  grams,  the  bifilar  suspension 

=m.  long  and  the  threads  about  0.05  cm.  apart.    Unfortunately  the  damp- 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.   107 

ing,  even  with  the  presence  of  a  water  well,  proved  insufficient  and  the  period 
too  long  in  the  case  of  the  suspension  used. 

An  example  of  the  six  groups  of  results  obtained  successively,  in  the  measure- 
ment of  one-fortieth  volt,  with  the  needle  charged  to  about  150  volts  may  be 
omitted,  the  mean  being  about  AAf= 0.028  cm.  or  A/V/AF=i.n  cm.  of 
micrometer  displacement  per  volt,  equivalent  to  27X10-°  volt  per  ring.  In 
spite  of  the  vertically  more  extended  needle,  there  has  therefore  been  no 
advantage  in  sensitiveness  over  the  preceding  case. 

The  following  experiments  were  made  at  different  voltages,  showing  better 
agreement  for  the  larger  voltages,  in  which  the  drift  is  less  significant,  rela- 


A7 

AJV 

AJV/AF 

Needle. 

Quadrant. 

volt 
0.0125 
•0375 
.0625 

cm. 
0.0135 
•045 
.077 

cm. 
i.  08 

1.20 

1.23 

// 

II 

tively.  Further  experiments  with  this  needle  led  to  no  new  results.  In  partic- 
ular the  endeavor  to  replace  the  silk  suspension  used  by  a  bifilar  the  threads 
of  which  were  single  fibers  of  silk,  proved  a  failure  owing  to  the  instability  of 
the  needle. 

72.  Further  observations.— The  same  needle  was  now  placed  within  large 
quadrants  (III),  n  cm.  in  diameter  and  2.3  cm.  high  within,  to  obviate  the 
difficulty  from  instability  in  a  needle  carrying  250  volts.  While  this  was 
accomplished,  the  drift  now  became  excessively  large.  The  mean  results  were 
AF  =  o.oi25  v0^;  AA/"  =  0.076  cm.;  AA/"/AV=6.i  cm.;  or  about  5X10-'  volt 
per  vanishing  ring.  Unfortunately  this  large  sensitiveness,  the  largest  obtained, 
could  not  be  controlled. 

It  appears  from  these  results  that  in  the  above  cases  the  actual  restoring 
torque  could  not  have  been  the  torsion  of  the  bifilar,  but  rather  a  directed 
residual  electrical  attraction  between  the  needle  and  the  quadrants,  the  torque 
of  the  fiber  being  operative  merely  in  placing  the  needle  in  the  fiducial  position, 
symmetrically  with  respect  to  the  division  line  between  the  quadrants.  In 
other  words,  the  displacement  of  the  needle  is  not  to  be  estimated  in  terms  of 
the  rate  at  which  the  bifilar  torque  changes  per  degree,  but  in  terms  of  a  very 
much  larger  coefficient  of  the  electrical  forces  in  question,  so  that  apart  from 
giving  position  to  the  needle  as  specified,  the  bifilar  acts  not  very  differently 
from  a  unifilar  suspension.  The  instrument  is  thus  much  less  sensitive  than 
would  be  inferred  from  the  dimensions  of  the  bifilar.  Hence  it  appeared  de- 
sirable to  return  to  the  needle  and  quadrants  in  §  70,  with  the  object  of  ascer- 
taining whether  the  sensitiveness  might  not  be  actually  increased  by  decreasing 
the  potential  of  the  needle  until  a  deflection  fully  corresponding  to  the  torque 
of  the  bifilar  should  show  itself.  The  present  point  of  view  also  indicates 
why  nothing  was  gained  by  the  use  of  a  larger  needle  in  §71,  seeing  that  in 
such  a  case  the  electrical  restoring  forces  increase  at  the  same  rate  as  the 


108  EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


deflecting  forces  arising  in  the  difference  of  potential  of  the  quadrants.  In 
the  same  way  the  negative  effect,  as  regards  sensitiveness,  of  an  increase  of 
the  potential  of  the  needle  above  a  certain  value  is  accounted  for.  Table  13 
contains  data  bearing  on  this  inference. 

TABLE  13. — Needle  II;  quadrants  II;  steel  frame. 


Needle  at— 

AF 

AN 

AJV/A7 

I    126  volts 

volt 
o.os 

cm. 
0.032 

cm. 
0.64 

II  150  volts 

.os 

.040 

.80 

III.  183  volts  

.025 

.009 

.36 

.05 

•075 

.100 

.025 
.047 
.062 

•50 
•63 
.62 

Thus  the  sensitiveness  rapidly  reaches  a  maximum  when  the  potential  of 
the  needle  is  about  150  volts,  after  which  it  more  gradually  diminishes  (see 
fig.  66,  curve  6). 

Furthermore,  in  Series  III,  where  the  needle  is  at  the  highest  potential 
applied,  the  sensitiveness  seems  to  increase  with  the  voltage  measured.  This, 
however,  is  merely  the  result  of  the  fact  that  there  is  apparently  a  small  fixed 
voltaic  potential  difference  between  the  quadrants,  even  if  they  are  nominally 
identical  or  in  the  connections.  Thus  in  fig.  66,  curve  a,  AF  and  A7V  are  pro- 


FIG.  66. 

portional  within  the  inevitable  errors;  but  the  deflections  begin  with  a  differ- 
ence of  potential  of  about  0.012  volt.  In  measuring  such  small  voltages 
electrostatically  these  voltaic  differences  become  of  serious  moment. 

The  maximum  sensitiveness  obtained  is  not  as  large  as  above,  being  but 
4oXio-«  volt  per  vanishing  ring.  Finally,  the  water  damper  was  removed, 

'  that  the  needle  was  subject  to  air  damping  only.  After  a  long  trial  it  was 
necessary  to  abandon  the  work,  as,  in  consequence  of  the  excessive  drift, 
measurement  was  out  of  the  question.  Most  of  this  drift  is  probably  intro- 
duced by  the  steel  frame. 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.   109 


73.  Observations,  continued.— The  framework  of  the  needles  above  was  of 
steel.  It  was  supposed  that  even  if  not  originally  magnetic,  such  a  needle 
might  be  subject  to  variations  of  the  earth's  field,  through  which  it  becomes 
temporarily  magnetic  by  induction.  Accordingly  a  needle  of  the  same  dimen- 
sions as  the  preceding  was  constructed  on  a  frame  of  thin  copper  wire  and 
tested,  with  the  results  of  table  14. 

TABLE  14.— Needle  II,  copper  frame.    Quadrants  II. 


Needle  at— 

AF 

AW 

AW/A  7 

140  volts  

volt 
0.0125 
.025 
.050 

•075 
.100 

cm. 
0.008 

.012 
.020 

•035 
.044 

cm. 
0.64 
.48 
.40 
•47 
•44 

The  sensitiveness  is  on  the  average  70  microvolts  per  ring,  a  smaller  value 
than  in  the  last  experiments,  owing  to  a  somewhat  greater  weight  of  the  needle. 
The  voltage  was  now  increased  further  and  the  following  experiments  made: 

TABLE  14. — Continued. 


Needle  at— 

AF 

AW 

AAVAF 

1  80 

o  025 

O  OI7 

O  7O 

2  SO 

•050 
•075 

.100 

•125 

.187 

OTO 

.025 
.046 
•053 
.071 
.098 
025 

•50 
.61 

•53 
•57 
•52 

CQ 

•075 

.100 

•033 
.041 

•44 

.41 

As  before,  the  sensitiveness  passes  through  a  maximum  when  the  voltage 
of  the  needle  is  about  180,  and  is  about  as  great  for  the  voltage  of  140  volts 
as  for  250  volts  (see  curve  /,  fig.  66).  The  maximum  sensitiveness  is  53 
microvolts  per  ring.  Though  the  drift  was  not  quite  removed,  the  stability 
of  the  needle  under  any  given  circumstances  proved  in  fact  to  be  greater  than 
before,  indicating  a  marked  improvement  as  the  result  of  replacing  the  steel 
frame  by  one  of  copper.  The  curves  c,  d  (raised  0.05  cm.),  e  (raised  o.i  cm.), 
show  that  AF  and  AN  are  proportional  within  the  limits  of  error.  The  latter, 
e,  seems  to  begin  with  an  initial  potential  which  would  mean  that  the  sensi- 
tiveness is  even  lower  at  250  volts  than  at  140  volts. 

It  therefore  seemed  necessary  to  replace  the  needle  in  some  other  of  the 
above  experiments  by  structures  not  containing  steel.  Thus  the  needle  and 
quadrants  used  in  §  71  with  this  improvement  gave  the  results  shown  in  table  15. 

In  view  of  the  low  voltage  of  the  needle,  the  sensitiveness  attained  is,  as 
above,  exceptionally  high.  The  displacement  AJV  is  proportional  to  AF  (see 
fig-  66,  g),  but  begins  with  a  permanent  potential  of  0.008  volt.  Something 
similar  to  this  occurs  in  some  of  the  above  results  on  a  smaller  scale,  so  that 


110  EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


it  is  not  impossible  that  voltaic  potential  differences  in  the  quadrants  (or 
connections)  may  be  in  question.  These  were  of  brass  and  nominally  identical ; 
but  a  difference  of  o.oi  volt  is  not  out  of  the  question.  Allowing  for  the  initial 
potential  difference,  the  sensitiveness  is  about  AW/A  V=  2.0  cm.  per  volt  or 
50  microvolts  per  vanishing  interference  ring.  The  needle  was  far  more  steady 
than  in  the  above  cases,  so  that  measurements  could  be  made  with  reasonable 
assurance.  A  curious  result  is  thus  attained:  on  widening  the  quadrants,  so 
that  the  distance  between  the  quadrants  and  the  needle  is  increased  within 
limits,  greater  sensitiveness  is  secured.  The  reason  has  been  suggested,  that 
inasmuch  as  the  electric  forces  which  place  the  needle  are  now  small,  the  latter 
is  subject  to  the  force  of  the  bifilar  suspension  only. 

TABLE  15. — Large  needle  III  in  large  quadrants  III.    Copper  frame. 


Needle  at— 

AF 

AJV 

AJV/A7 

185  volts  

volt 
0.009 
.018 
.036 

cm. 
0-035 
•055 
•093 

3-8 
3-0 

2.6 

Further  work  was  done  with  the  needle  at  250  volts;  but  an  adequately 
stable  condition  of  the  needle  could  not  be  obtained,  as  it  gradually  crept 
beyond  the  range  of  the  interferometer. 

Finally,  experiments  were  made  with  a  needle  of  the  ordinary  form  (I, 
§  69),  inclosed  in  the  intermediate  quadrants  (II).  The  relatively  sharp  edges 
of  the  needle  should  reduce  the  electric  torque. 

TABLE  1 6.— Needle  I.    Quadrants  II.    Copper  frame. 


Needle  at— 

A7 

AN 

AJV/A7 

145  volts  

volt 
0.018 
.072 

cm. 

O.OIO 

•053 

0.56 
•74 

The  sensitiveness  here  is  not  inferior  to  the  usual  cases  above,  being  on  the 
average  46  microvolts  per  ring,  and  this  in  conformity  with  the  relatively  low 
potential  of  the  needle.  The  following  results  were  obtained  at  higher  poten- 
tials with  the  same  needle: 


TABLE  16.— Continued. 


Needle  at— 

A7 

AAT 

Atf/AF 

240  

o  018 

•036 
.072 
.108 

•145 

.019 
•035 
•051 

unstable. 

-53 
•50 

•47 

In  spite  of  the  much  larger  potential  of  the  needle  in  the  last  series,  the 

average  sensitiveness  is  again  less,  showing  the  same  behavior  as  the  above 

ses.    The  relation  of  potential  and  displacement  is  linear  (fig.  66,  curve  fc), 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.   Ill 


but  the  displacements  begin  with  a  potential  of  0.009  volt.  Allowing  for  this, 
the  sensitiveness  is  66  microvolts  per  vanishing  interference  ring.  The  steadi- 
ness of  the  needle  in  both  series  of  experiments  was  exceptional  for  this  labora- 
tory, so  that  the  motion  of  single  rings  could  at  times  have  been  counted. 
Nevertheless  the  relative  values  of  successive  displacements  for  the  same 
potential  increment  were  not  superior  to  the  above. 

Since  di=dd/2a  sin  i,  the  electrometer  itself  in  the  most  sensitive  detailed 
case  (needle  III,  quadrant  III)  was  only  moderately  sensitive,  for  if  1  =  45°, 
0  =  4.5  cm.,  dd/AV=&N/AV=2  or  A*/A7=o.3a  radian  per  volt  =  i8.4°  per 
volt.  Hence,  the  reflected  ray  in  the  ordinary  mirror  and  scale  adjustment, 
at  i  meter  distance  of  scale  from  mirror,  would  move  over  about  64  cm.  per 
volt.  In  one  of  the  incidental  cases  above,  it  is  true,  about  three  times  this 
value  was  reached.  In  the  other  cases  it  was  proportionately  less  sensitive. 
Thus  for  AAT/A  V = o.  5 ,  the  deflection  would  be  but  1 6  cm.  per  volt.  No  doubt 
much  could  be  accomplished  by  making  the  electrometer  itself  more  sensitive; 
but  this  improvement  was  not  the  immediate  purpose  of  the  present  article. 

Other  comparative  experiments  with  copper-framed  needles  were  now  made. 
The  sharp-edged  needle,  I,  placed  in  the  large  quadrants  III  gave  the  results 
of  table  17. 

TABLE  17. — Needle  I.    Quadrants  III.    Copper  frame. 


Needle  at— 

A7 

AN 

Atf/A7 

2  SO 

volt 
0.018 

cm. 

O.OIO 

O  S6 

.036 
.072 
.108 
.144 

.016 
.040 

.081 
.115 

11 
fi 

In  this  case  the  displacements  are  not  proportional  to  the  voltages,  but 
increase  at  an  accelerated  rate.  Neither  do  they  seem  to  begin  at  the  origin. 
The  sensitiveness  accordingly  increases  rapidly  with  increased  deflection,  but 
its  mean  value  is  of  the  ordinary  magnitude.  This  behavior  of  a  thin  needle, 
in  relatively  wide  quadrants,  where  stability  should  have  been  insured,  was 
unexpected;  but  by  raising  the  needle  the  following  results  were  found,  showing 
that  the  needle  above  was  inadequately  centered: 

TABLE  17. — Continued. 


Needle  at— 

AF 

AJV 

AW/AF 

2  CO 

0.018 

0.023 

1.3 

.036 
.072 
.108 

.038 

•075 
.in 

i.  06 
1.04 
1.03 

This  result  is  a  great  improvement;  for  not  only  is  the  potential  proportional 
to  the  displacement  (see  curve  i,  fig.  66),  but  the  sensitiveness  is  much  larger 
than  heretofore,  for  the  same  needle,  being  about  30  microvolts  per  vanishing 


112  EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER. 


ring.  The  advantage  of  wide  quadrants  is  thus  again  sustained.  The  sensi- 
tiveness, however,  lags  behind  the  result  for  the  large  biplane  needle  III 
(curve  g)  under  the  same  circumstances. 

The  intermediate  biplane  needle  II  in  the  quadrants  III  shows  the  results 
of  table  18. 

TABLE  18.— Needle  II.    Quadrants  III.    Copper  frame. 


Needle  at— 

A7 

AN 

&N/&V 

volt 

cm. 

0.018 

0.007 

0.39 

•036 

.022 

.61 

.072 

.050 

.70 

.108 
.144 

.068 
•095 

A 

The  sensitiveness  is  low,  owing  again,  no  doubt,  to  the  position  of  the 
needle.  Otherwise  the  observations  are  good.  The  needle  was  then  raised, 
with  the  following  results: 

TABLE  18. — Continued. 


Needle  at— 

A7 

Atf 

AW/A  7 

250  

0.036 
.072 
.108 
.144 
.018 

0.023 
.050 
.078 
.103 
.015 

0.64 
.69 
.72 

£ 

The  sensitiveness  has  been  slightly  increased.  Moreover,  it  grows  larger  in 
the  course  of  the  work,  as  if  some  surface  viscosity  in  the  liquid  of  the  damper 
were  gradually  overcome.  The  instrument  in  general  behaved  admirably, 
barring  alone  the  presence  of  drift  which  can  not  in  the  present  laboratory  be 
quite  overcome.  As  the  fibers  were  but  13.5  cm.  long  as  compared  with  23.0 
cm.  above,  the  mean  reduced  sensitiveness  was  25  microvolts  per  ring.  It 
is  thus  inferior  both  to  the  large  biplane  and  to  the  wedge,  for  reasons  which 
do  not  appear.  The  sensitiveness  should  have  been  intermediate. 

Finally,  the  intermediate  quadrants  II  were  again  mounted  with  the  same 
needle,  the  quadrants  being  specially  smoothed  inside,  so  as  possibly  to  elimi- 
nate electric  restoring  forces.  The  results,  however,  were  not  essentially 
different  from  the  above. 

74.  Summary. — The  results  of  this  long  and  excessively  laborious  paper 
may  be  given  in  a  few  words.  By  providing  the  needle  of  the  quadrant  elec- 
trometer with  a  pair  of  mirrors,  in  parallel,  and  observing  displacements  on 
the  interferometer,  voltages  as  small  as  10  microvolts  may  be  detected  per 
vanishing  interference  ring,  so  that  a  single  microvolt  should  be  reached  by 
estimation.  In  the  above  experiments  this  could  not  be  done,  because  the 
needle  was  never  confined  to  a  fixed  position  of  equilibrium,  to  an  extent  com- 
patible with  the  use  of  light-waves.  The  causes  of  this  drift  are  incidental, 


EXPERIMENTS  WITH  THE  DISPLACEMENT  INTERFEROMETER.   113 

probably  attributable  to  air-currents,  convection  currents  due  to  temperature 
differences  and  pendulum  motion  of  the  needle  resulting  from  tremors.  Steel 
must  always  be  excluded  from  the  framework  of  the  needle. 

The  sensitiveness  as  is  otherwise  known,  theoretically,  does  not  in  any  case 
increase  with  the  potential  of  the  needle,  but  passes  through  a  maximum  (in 
the  above  designs)  usually  at  about  150  volts.  This  is  the  case  both  with 
the  sharp-edged  and  the  cylindrically-faced  biplane  needles.  The  directing 
force  in  the  case  of  such  needles  is  essentially  electric;  i.e.,  they  are  set  in  a 
position  of  equilibrium  relatively  to  the  quadrants  by  electric  stress  large  in 
comparison  with  the  torque  of  the  bifilar.  As  soon  as  these  forces  increase  at 
the  same  rate  as  the  potential  of  the  needle,  the  further  increase  of  the  latter 
is  no  longer  serviceable.  Hence  the  biplane  needle,  set  in  relatively  wide 
quadrants,  was  found  to  offer  the  best  conditions  of  sensitiveness,  and  it  is 
in  the  case  of  needles  and  quadrants  of  this  design  that  the  best  results  were 
obtained.  In  other  words,  the  sensitiveness  also  passes  through  a  maximum 
as  the  mean  distance  between  the  outside  contours  of  the  needle  and  the  inside 
contours  of  the  quadrants  increases. 

After  preliminary  experiments,  the  optics  of  the  instrument  offered  no 
serious  difficulty.  It  is  merely  necessary  to  follow  the  reflected  light  by  placing 
white  screens  behind  each  mirror  in  the  direction  of  the  impinging  rays.  Since 
the  rays  are  reflected  at  the  grating,  the  returning  ray  also  necessarily  passes 
through  the  grating,  and  this  part  of  the  adjustment  is  therefore  automatic. 
With  a  copper-framed  needle,  the  water  damper  will  probably  not  be  essential, 
in  which  case  the  discrepancies  due  to  surface  viscosity  will  also  disappear. 


43 


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